2.1 $P_4$ and fixing operations

The case of $P_4$ -free graphs is particularly interesting as there is no finite $P_4$ -induced-saturated graph.

We find an infinite $P_4$ -induced-saturated graph G by considering a sequence $(G_i)_{i\in \mathbb {N}}$ of graphs and taking its limit. We define the sequence $(G_i)$ inductively. Throughout the construction, we keep a list of pairs which are “bad,” in the sense that perturbing them does not create a copy of $P_4$ . We obtain $G_{i+1}$ from $G_i$ by attaching a gadget that “fixes” the next bad pair in the list without creating a copy of $P_4$ . We then update the list by adding the bad pairs involving a new vertex to the end of the list.

When there is a simple fixing operation for H, this gives us a straightforward recipe for proving Corollary 1.2. In the case of $P_4$ , the construction is as follows. Given a bad pair $\{x,y\}$ , we create twinsFootnote ² $x'$ and $y'$ of x and y. Since $P_4$ has no twins, adding a twin can never create an induced copy of $P_4$ . Moreover, it is not hard to see that in the resulting graph perturbing the pair $\{x,y\}$ creates a copy of $P_4$ . The sequence $(G_i)_{i\in \mathbb {N}}$ of graphs is obtained by considering an arbitrary $P_4$ -free graph, computing a list of its bad pairs, then obtaining $G_{i+1}$ from $G_i$ by fixing its first bad pair and updating the list of bad pairs.

We give an example in Figure 2: we begin with a graph $G_1$ with a single bad pair $\{u_1,u_2\}$ , and fix it to obtain $G_2$ with two bad pairs $\{\{u_1,v_1\},\{u_2,v_2\}\}$ . Fixing $\{u_1, v_1\}$ creates a graph $G_3$ with three bad pairs, with $\{u_2,v_2\}$ at the front of the list.

Figure 2:

A sequence of graphs $G_1,G_2,G_3,\dots $ is obtained by repeatedly applying a fixing operation. The red edges and red non-edges are not fixed.

We note that, for any i, and for any bad pair $\{x,y\}$ of a graph $G_i$ , there is a (later) index j such that perturbing $\{x,y\}$ in $G_j$ or any subsequent graph yields an induced $P_4$ . In other words, any bad pair is ultimately fixed. Therefore, there is a countably infinite $P_4$ -free graph $G_{P_4}$ such that perturbing any pair in $G_{P_4}$ results in an induced copy of $P_4$ . That is, Corollary 1.2 is true for $H=P_4$ and, more generally, for any graph H that admits fixing operations for bad pairs.

2.2 $C_5$ and gatekeepers

The fixing operations for $P_4$ defined above rely heavily on the strong structural properties of $P_4$ -free graphs. Since we need to prove Corollary 1.2 for every graph H that is not a clique or independent set, we will need something much more general. The naïve way to define a fixing operation for H is by gluing a copy of $H-e$ with the missing edge aligned with a bad non-edge (or a copy of $H+e$ with the extra edge aligned with a bad edge). While this guarantees that perturbing the bad pair will result in an induced copy of H, the pitfall is that the gluing might create a copy of H. One may hope to avoid this by choosing carefully which edge to add or delete in H, but this may not always be possible (consider $H = P_4$ ).

However, for $H=C_5$ , this is a good approach (see Figure 3). To fix a bad non-edge $xy$ , we may glue $C_5-e$ by adding three vertices $u_1,u_2,u_3$ and four edges $xu_1,u_1u_2,u_2u_3,u_3y$ . Note that the resulting graph is $C_5$ -free, as every induced copy of $C_5$ involving some new edge would involve all of them. Similarly, to fix a bad edge $xy$ , we may glue $C_5+e$ by adding three vertices $u_1,u_2,u_3$ and five edges $xu_1,u_1y,xu_2,u_2u_3,u_3y$ . It is not hard to see that the resulting graph is also $C_5$ -free. In fact, this idea works for many graphs, including every $3$ -connected graph H that is not a clique. Indeed, if we “fix” the pair $\{x,y\}$ in a similar fashion to Figure 3, then $\{x,y\}$ forms a 2-cut in the resulting graph between the “old” and “new” vertices. If H is 3-connected, then it cannot contain vertices of both types so no new copies of H can be created. This proves Corollary 1.2 for any $3$ -connected graph H.

Figure 3:

The fixing operation for non-edges (left) and edges (right) is depicted. The blue vertices represent newly created vertices.

We now formalize what makes $C_5$ and 3-connected graphs (but not $P_4$ ) behave well. An edge $uv$ in a graph H is a gatekeeper if for any H-free graph G, gluing a copy of $H - uv$ on a non-edge of G results in an H-free graph. A similar definition holds for non-edges. We say that H admits a gatekeeper of each type if it contains both an edge and a non-edge that are gatekeepers.

A graph H that admits a gatekeeper of each type admits a fixing operating for bad edges as well as for bad non-edges, using the naïve approach described above. Note that a path contains no edge that is a gatekeeper.

2.3 $C_5$ with a leaf and cores

Unfortunately, even small modifications to a graph can make a graph which is much harder to handle. For example, consider the graph obtained from $C_5$ by adding a pendant edge to one of the vertices. While we easily found fixing operations for $C_5$ using gatekeepers, this new graph does not admit gatekeepers of either type. Instead, we focus on the $C_5$ and work within the class of $C_5$ -free graphs. If we work within this class, we can use the gatekeepers for $C_5$ and not create a copy of $C_5$ or of our target graph. These gatekeepers are not quite enough to guarantee a perturbation creates a copy of $C_5$ plus a leaf, but this is easily solved by adding a leaf to every vertex we add. Note that the leaves cannot be in a $C_5$ , so this is a valid modification. We now formalize and extend this idea using the notion of cores.

The $2$ -core $H^*$ of a finite graph H is obtained by iteratively removing vertices of degree at most 1. Crucially, if we take a graph G which is $H^*$ -free, then adding a leaf to each vertex of G does not create a copy of $H^*$ let alone H, nor does adding an isolated vertex. Given a fixing operation for $H^*$ , we can create a fixing operation for H as follows. Throughout the process, we maintain that the graph created so far is $H^*$ -free. We interchange fixing operations for $H^*$ with steps that either add a leaf to each vertex or add an isolated vertex. Since every bad pair is eventually fixed by the fixing operation for $H^*$ , if we perturb a pair then we can embed a copy of $H^*$ . We can extend our copy of $H^*$ into a copy of H as we have added infinitely many leaves and isolated vertices.

Therefore, if a graph $H^*$ admits fixing operations, any graph H whose $2$ -core is $H^*$ admits fixing operations. We can generalize this notion by iteratively deleting any vertex with at most k neighbors or at most $\ell $ non-neighbors.

In particular, the 3-core $H^*$ of a graph H is obtained by iteratively removing vertices of degree at most 2. By the discussion above, Corollary 1.2 holds for a graph H if its 3-core is 3-connected. We are also done when the complement $\overline {H}$ of H is 3-connected: the complement of an $\overline {H}$ -induced-saturated graph is H-induced-saturated. When H has a 2-cut and both “sides” contain at least three vertices, then $\overline {H}$ is close to having a 3-connected 3-core. However, there is one “bad” 2-cut to take into account, depicted in Figure 4.

Figure 4:

A “bad” 2-cut is shown. No such cuts are present in the $3^*$ -core.

We define the $3^*$ -core of a graph H as the graph obtained from H by iteratively removing vertices of degree at most $2$ , as well as true twins of degree $3$ (“bad” 2-cuts). As before, fixing operations for a graph $H^*$ can be extended to any graph H whose $3^*$ -core is $H^*$ . If a graph H has a non-empty $3^*$ -core, then either the $3^*$ -core of H or the $3^*$ -core of $\overline {H}$ is 3-connected. There is a small catch: the $3^*$ -core could be a clique and this will give a few more cases to handle.

2.4 Outline of the proof

We can now give a broad overview of the steps in our proof. The proof will be given formally in Section 7.6.

• • We give a fixing operation for all H whose $3^*$ -core is 3-connected and not a clique. The reader should already be able to verify this for the weaker Corollary 1.2 from the discussion above. We provide the stronger statements needed for Theorem 1.1 in Section 4.

• • We give a fixing operation for all H whose 2-core is a $K_{1,1,p}$ or $K_{2,p}$ and a direct construction for all forests with a unique vertex of maximum degree in Section 6.

• • We prove a structure theorem (Theorem 5.1) that shows that for all finite graphs H on at least 12 vertices, either H or its complement falls into one of the above categories. This is done in Section 5.

• • We provide direct constructions for specific graphs H on at most seven vertices in Section 7.

• • We use a computer to check that for all graphs H on at most 11 vertices, either H or its complement falls into one of the above categories. This is detailed in Section 7.3.

2.5 From individual perturbations to locally finite perturbations

For Corollary 1.2, we only need to generate a copy of H when we make a single perturbation: we either turn an edge into a non-edge or vice versa. For Theorem 1.1, we allow many pairs to be perturbed at once, and the arguments above no longer directly apply.

We describe one way in which we show that a pair has been fixed. After a locally finite perturbation, we want to embed H vertex-by-vertex. Suppose that we already embedded $h_1,\dots ,h_s$ . If there are infinitely many options from which to choose the next vertex $h_{s+1}$ , then there is always an option for which the adjacencies to $h_1,\dots ,h_s$ have not been altered by the locally finite perturbation. Indeed, for each $h_i$ , there are only finitely many v so that $\{h_i,v\}$ has been altered. One way to generate infinitely many options is to blow up vertices into infinite cliques. This does not always work, since it may create a copy of H. Moreover, it also creates new edges to be “fixed.” Nevertheless, the difference between allowing a single perturbation or allowing any locally finite perturbation is reasonably small in most of our proofs. In some places, it does require additional insights, such as for 3-connected graphs (handled by Lemma 4.1).

There is one small additional point that comes up when we prove our stronger variant. Assume that a graph H admits fixing operations as in Section 2.1. For Corollary 1.2, we start from a finite graph, and fixing a pair adds finitely many vertices, hence finitely many new bad pairs. Each graph of the sequence is finite, and new bad pairs can simply be added to the end of the list of existing bad pairs. However, in the proof of Theorem 1.1, we add countably infinitely many vertices to fix a single bad pair, thus possibly infinitely many new bad pairs. Since we add countably many bad pairs for a countable number of times, it is possible to schedule the fixes in a careful order by a standard enumeration argument; this is described in the proof of Lemma 6.4.

3.1 Locally finite perturbation

For a pair of distinct vertices $u,v\in V(G)$ , perturbing the pair $uv$ is the operation which removes $uv$ from $E(G)$ if it is present and adds it if it is not present. A locally finite perturbation of G is a graph obtained from G by perturbing arbitrarily many pairs (at least one, possibly infinitely many) under the constraint that for any vertex v, the number of perturbed pairs involving v is finite.

3.2 Strongly saturating

For a finite graph H, say a graph G is strongly H-induced-saturated if G does not contain H as an induced subgraph, but any locally finite perturbation of G does. Note that the very notion of strongly H-induced-saturated requires G to be infinite. We say that G is H-induced-saturated if G does not contain H as an induced subgraph, but a copy of H is created when an edge is added to G or removed from G.

3.3 Fixed and unfixed edges

Say a pair $xy$ in a graph G is unfixed (for H) if there is a locally finite perturbation which perturbs the pair $xy$ and does not result in an induced copy of H. Otherwise, we call the pair fixed (for H). Note that a graph G is strongly H-induced-saturated if and only if all pairs of G are fixed for H.

3.4 Gluing

Given two graphs G and $G'$ on disjoint vertex sets with $A\subseteq V(G)$ and $A' \subseteq V(G')$ , and a bijection $f:A'\to A$ , the graph $G"$ obtained from gluing $G'$ on G along f has $V(G")=V(G)\cup V(G')\setminus A'$ and $uv\in E(G")$ for distinct $u,v\in V(G")$ if and only if

1. (1) $uv\in E(G)$ ,

2. (2) $uv\in E(G')$ ,

3. (3) $u\in A$ and $f^{-1}(u)v\in E(G')$ , or

4. (4) $u,v\in A$ and $f^{-1}(u)f^{-1}(v)\in E(G')$ .

We will most commonly apply this operation for $A=\{u,v\}$ , $A'=\{u',v'\}$ and $f(u)=u',f(v)=v'$ . An example of this is given in Figure 5.

Figure 5:

An example is given on how two graphs can be glued on non-edges $\{u,v\}$ and $\{u',v'\}$ .

3.5 Fixing operations

We will sometimes need to maintain the stronger property that the graph is not just H-free but that it also does not contain a copy of some core of H. For this reason, we define a fixing operation relative to a class of graphs, as follows.

Given a class of graphs $\mathcal {F}$ , we say that the finite graph H admits an edge fixing operation (resp. non-edge fixing operation) for $\mathcal {F}$ if for every $G\in \mathcal {F}$ and every edge (resp. non-edge) $xy$ of G, there exists a graph $G'$ obtained from gluing a graph onto G such that

• • $G'\in \mathcal {F}$ , and

• • any locally finite perturbation of $G'$ which perturbs $xy$ contains a copy of H.

Crucially, G is an induced subgraph of $G'$ . We say that H admits a fixing operation for $\mathcal {F}$ if it admits both an edge fixing operation and a non-edge fixing operation. We show in Lemma 6.4 that if H admits a fixing operation for a non-empty class $\mathcal {F}$ of H-free graphs, then there exists a strongly H-induced-saturated graph.

3.6 Gatekeepers

An edge (resp. non-edge) $uv$ in a graph H is a gatekeeper if for any H-free graph G, gluing a copy of $H - uv$ on a non-edge (resp. edge) of G results in an H-free graph.

3.7 Cores

Given a graph H, the $(k, \ell )$ -core is the graph obtained by iteratively removing vertices which have fewer than k neighbors or less than $\ell $ non-neighbors. We write k-core to refer to the $(k,0)$ -core.

The $3^*$ -core of a graph H is obtained by iteratively removing vertices which have at most two neighbors and removing pairs $uv$ of vertices with $|N[u]\cup N[v]|\leq 4$ .

6.1 Fixing operations for $K_{2,p}$

The fixing operation for $K_{2,p}$ is relatively simple.

6.2 Fixing operation for $K_{1,1,p}$

The edge fixing operation for $K_{1,1,p}$ is more involved and requires an additional assumption about the graphs it is applied to.

6.3 Forests with a vertex of unique maximum degree

Let F be a finite forest which has a unique vertex of maximum degree $d>1$ that we denote by v. One easy way of preventing a graph G from containing a copy of F is to ensure it has maximum degree strictly less than d. Unfortunately, removing an edge from G is not going to increase the maximum degree and create a copy of F, so we need to be slightly smarter. Since F is a forest, the neighborhood of any vertex is an independent set and, instead of simply restricting the maximum degree, we can ensure that no vertex in G has a neighborhood containing an independent set of size d.

We define the graph $K_\infty ^p$ as the infinite graph with vertex set $\mathbb {Z}^p$ and an edge between a vertex $u = (u_1, \dots , u_p)$ and vertex $v = (v_1, \dots , v_p)$ if and only if they disagree in exactly one coordinate, that is, there is a unique i such that $u_i \neq v_i$ . The maximum degree is unbounded, but the neighborhood of any vertex can be decomposed into p cliques, one for each coordinate, and the largest independent set in the neighborhood of any vertex is of size p. We claim that $K_\infty ^{d-1}$ is strongly F-induced-saturated.

Proof As argued above, $K_\infty ^{d-1}$ does not contain F since the largest independent set in any neighborhood is smaller than the maximum degree in F. It remains to show that making any locally finite edit creates an induced copy of F.

We first consider the case in which F is a tree. Let v be the unique vertex of maximum degree and suppose we remove an edge. Without loss of generality, let the edge be from $(1, 0, \dots , 0)$ to $(2, 0, \dots , 0)$ . There must exist an integer $i_v$ such that $(i_v,0,\dots ,0)$ is still connected to both $(1,0,\dots ,0)$ and $(2,0,\dots ,0)$ . Here, we will embed our vertex v. Similarly, there are integers $j_3, \dots ,j_d$ such that after the locally finite edit, there are still edges from $(i_v,0,\dots ,0)$ to

$$\begin{align*}(i_v,j_3,0,\dots,0),(i_v,0,j_4,\dots,0),\dots,(i_v,0,0,\dots,j_d) \end{align*}$$

and the vertices displayed above together with $(1,0,\dots ,0),(2,0,\dots ,0)$ form an independent set. We embed the d neighbors of v into this independent set of size d.

We now explore the tree, heading out from v and assigning each vertex to a vertex in $\mathbb {Z}^{d-1}$ as we go. Suppose we have reached the vertex $u = (u_1, \dots , u_{d-1})$ which disagrees with the previous vertex in coordinate i. There are at most $d' \leq d-1$ neighbors of u, say $w_0, w_1, \dots , w_{d'-1}$ , and we have already seen one of them, say $w_0$ . We must assign to each $w_j$ with $j\in [d]$ a vertex in $\mathbb {Z}^{d-1}$ . First choose $d'-1$ distinct coordinates $p_1, \dots , p_{d'-1}$ , none of which are equal to i, and set $w_j$ to be equal to u except in position $p_j$ . For position $p_j$ , we choose an integer which ensures that $w_j$ is not adjacent to any of the vertices we have embedded except for u. Since at each step, there are only finitely many “bad integers” to choose out of infinitely many, we can always select an appropriate one.

Adding an edge can be handled similarly: we embed v into an endpoint x of the edge and note that the added edge again makes it possible to embed the d neighbors of v as an independent set in the neighborhood of x.

For arbitrary forests F, let T be the connected component of F which contains the unique vertex of maximum degree. We showed above that any locally finite edit creates a copy of T. All other connected components of F have maximum degree at most $d-1$ . After embedding T, it is straightforward to embed the other components in $K_\infty ^{d-1}$ in a way that there are no edges to the vertices used for T.

6.4 Scheduling the fixes

The following lemma is straightforward but important. A key part of the proof is that any bad pair is fixed in a finite number of steps. This ensures that every bad pair is fixed in the limit.

Proof We define a sequence of graphs $G_1\subseteq G_2 \subseteq \dots $ in $\mathcal {F}$ and we define the graph G as $\cup _{i=1}^\infty G_i$ .

Let $G_1\in \mathcal {F}$ . We keep track of a set F of pairs that need to be fixed with a priority order (described later in this proof), which is initialized as the (countable) set of unfixed pairs in $G_1$ . Suppose we have defined $G_i\in \mathcal {F}$ for some integer $i\geq 1$ . If all pairs $xy\in G_i$ are fixed, we may set $G_t=G_i$ for all $t\geq i$ and $G=G_i$ is our desired graph. Otherwise, let $xy\in G_i$ be an unfixed pair of highest priority in F. We let $G_{i+1}$ be the graph obtained from applying the fixing operation for this pair.

Next, we describe how to define the priority order in such way that every bad pair is eventually fixed. We first fix an enumeration $<$ on $\mathbb {N}\times \mathbb {N}$ , for example,

$$\begin{align*}(1,1)<(1,2)<(2,1)<(2,2)<(3,1)<(3,2)<(3,3)<\dots \end{align*}$$

At any point of our sequence, $G_i$ is a countable graph and when created, we may enumerate the vertex pairs of $G_i$ . We prioritise the jth vertex pair of $G_i$ above the $j'$ th vertex pair of $G_{i'}$ if and only if $(i,j)<(i',j')$ . At any point, we perform next the fix of highest priority among those that involve a pair of vertices both existing in the current graph and for which the vertex pair is currently unfixed. (If desired, we can also allow to fix vertex pairs that are already fixed.)

By definition of a fixing operation for $xy$ , if $G_i\in \mathcal {F}$ , then $G_{i+1}\in \mathcal {F}$ and the pair $xy$ is fixed in $G_{i+1}$ . Moreover, any pair which was fixed in $G_i$ remains fixed in $G_{i+1}$ since $G_i$ is an induced subgraph of $G_{i+1}$ . By putting the priority order on F, we ensure that for any pair $xy$ in G, there is a finite i such that $xy$ is a fixed pair in $G_i$ , so in particular, $xy$ is a fixed pair in G.

Since no copy of H can be created in $G_i$ for all i, G is also H-free. This proves that G has the desired properties.

7.1 The up and right graph

We start with a construction which we call the up and right graph, which handles all graphs which are “close to” a permutation graph. A graph G on the vertices $\{v_1, \dots ,v_n\}$ is a permutation graph if there is a permutation $\sigma \in S_n$ such that, for every $i < j$ , the edge $v_iv_j$ is present if and only if $\sigma (i) < \sigma (j)$ . A graph is close to a permutation graph if it is not a permutation graph, but adding or removing any edge creates a permutation graph.

The up and right graph is the graph on $\mathbb {Q} \times \mathbb {Q}$ where the vertex $(p,q)$ is connected to $(s,t)$ if and only if one of the following holds:

• • $(q + r \sqrt 2) < (s + t \sqrt 2)$ and $(q - r \sqrt 2) < (s - t \sqrt 2).$

• • $(q + r \sqrt 2)> (s + t \sqrt 2)$ and $(q - r \sqrt 2)> (s - t \sqrt 2).$

That is, two vertices are connected if one of the vertices is up and right of the other after the linear transformation $(q,r) \mapsto (q + r \sqrt 2, q - r \sqrt 2)$ has been applied. Our motivation for this is to ensure that no two vertices can differ in exactly one coordinate. Indeed, if

$$\begin{align*}q \pm r \sqrt2 = s \pm t \sqrt2, \end{align*}$$

where $q,r,s,t \in \mathbb {Q}$ , then we must have $(q,r) = (s,t)$ . This means that given n distinct vertices $v_i = (q_i, r_i)$ for $i \in [n]$ , we can order them such that

$$\begin{align*}q_i + r_i \sqrt2 < q_j + r_j \sqrt2 \end{align*}$$

whenever $i < j$ .

Similarly, there is also an order based on the other coordinate, and there is a unique permutation $\sigma \in S_n$ such that

$$\begin{align*}q_{i} - r_{i} \sqrt2 < q_{j} - r_{j} \sqrt2 \end{align*}$$

whenever $\sigma (i) < \sigma (j)$ . With this notation, there is an edge between $v_i$ and $v_j$ , where $i < j$ , exactly when $\sigma (i) < \sigma (j)$ . This means that any induced subgraph is a permutation graph.

The following lemma shows that every permutation graph can be found as an induced subgraph of the up and right graph, giving us a characterization of exactly which finite graphs appear as induced subgraphs.

Proof We already argued that any induced subgraph is a permutation graph, so it only remains to show how to find a copy of any given permutation graph. Let $\sigma $ be a permutation that defines the permutation graph H and consider the n points

$$\begin{align*}\tilde{v}_i = \left( \frac{i + \sigma(i)}{2} , \frac{i - \sigma(i)}{2 \sqrt{2}} \right) \text{ for }i\in [n].\end{align*}$$

The coordinates are chosen so that

$$\begin{align*}\frac{i + \sigma(i)}{2}+\frac{i - \sigma(i)}{2 \sqrt{2}} \sqrt{2} = i \end{align*}$$

and

$$\begin{align*}\frac{i + \sigma(i)}{2}-\frac{i - \sigma(i)}{2 \sqrt{2}} \sqrt{2}=\sigma(i). \end{align*}$$

Ideally, we would take the $\tilde {v}_i$ as the vertices in the up and right graph, but the $\tilde {v}_i$ are not necessarily rational. However, the rational points are dense in $\mathbb {R}^2$ and we can choose vertices of the up and right graph which are sufficiently close to the points $\tilde {v}_i$ to keep the expected adjacencies. This provides a subgraph isomorphic to H.

Given this lemma is it not hard to prove the following theorem classifying the graphs which are strongly saturating in the up and right graph.

Proof It is easy to see that these conditions are necessary. Indeed, the up and right graph cannot contain a copy of H so, by Lemma 7.1, H is a not a permutation graph. Adding an edge e to the up and right graph must create a copy of H and e corresponds to some edge $e'$ of this copy of H. This means there is an induced copy of $H - e'$ in the up and right graph and $H - e'$ is a permutation graph. Similarly, there must be a non-edge $\overline {e}$ of H such that $H + \overline {e}$ is a permutation graph.

We now show that the conditions are sufficient. Since H is not a permutation graph, there is no copy of H in the up and right graph, and it remains to show that making any locally finite edit creates a copy of H. Let us first consider adding a single edge. By assumption, there is an edge $e = v_i v_j$ of H such that $H-e$ is a permutation graph, and we claim that this is enough to guarantee the existence of a copy of H after adding an edge.

Suppose $H-e$ is a permutation graph and let f be a non-edge from $(q,r)$ to $(s,t)$ in the up and right graph. The up and right graph is vertex transitive so we can assume that $(q,r) = (0,0)$ . We now repeat the construction from Lemma 7.1, but with the addition of extra scaling factors. Define

$$\begin{align*}k' = (k -i) \frac{s + t\sqrt2}{j - i} \text{ and } \sigma'(k) = (\sigma(k) - \sigma(i)) \frac{s - t\sqrt2}{\sigma(j) - \sigma(i)}, \end{align*}$$

and let

$$\begin{align*}\tilde{v}_k = \left( \frac{k' + \sigma'(k)}{2}, \frac{k' - \sigma'(k)}{2 \sqrt{2}}\right).\end{align*}$$

Note that by our choice of the scaling factors, we have $\tilde {v}_i = (0,0)$ and $\tilde {v}_j = (s,t)$ , so that e corresponds to f. Since e is not an edge in the permutation graph, $j - i$ and $\sigma (j) - \sigma (i)$ have opposite signs. Likewise, since f is not an edge in the up and right graph, ${s + t\sqrt 2}$ and ${s - t\sqrt {2}}$ also have opposite signs, and the scaling factors both have the same sign. Hence, if the $\tilde {v}_k$ were vertices in the up and right graph, we would have found an induced subgraph isomorphic to $H - e$ and we would be done. Instead, we may need to perturb the $\tilde {v}_k$ by a small amount for $k \neq i, j$ to make sure that they are rational.

Using an almost identical argument, one can show that removing an edge creates a copy of H provided that there is a non-edge $\overline {e}$ of H such that $H + \overline {e}$ is a permutation graph.

So far, we have only shown that the graph is induced-saturated for H and contains a copy of H when a single perturbation is made, but it is not hard to adapt the proof to handle locally finite perturbations. We begin as before, setting $v_i$ and $v_j$ as $\tilde {v}_i$ and $\tilde {v}_j$ , respectively. We then choose each $v_k$ in turn. At each step, there are infinitely many choices for $v_k$ that are sufficiently close to $\tilde {v}_k$ . Since the edit is locally finite and we have only picked finitely many vertices so far, we can choose such a vertex (in fact, all but finitely many work) for which none of the adjacencies to the vertices we have already chosen have been altered.

7.2 The torero graph

In this section, we give a construction which is strongly-H-induced-saturated whenever the $(1,1)$ -core is a copy of the bull graph (see Figure 6) or $P_4$ .

Figure 6:

The bull graph.

The torero graph has vertex set $\mathbb {Q} \cap (0,1)$ and there is an edge from x to y if and only if $x + y> 1$ .

Proof We prove this theorem in three parts. First, we show that there is no induced copy of $P_4$ in the torero graph (and hence no copy of the bull graph). Then we show that any locally finite modification of the torero graph contains an induced copy of the bull graph (and hence an induced copy of $P_4$ ). Finally, we observe that the torero graph is strongly-H-induced-saturated when the $(1,1)$ -core is a copy of the bull graph or $P_4$ .

Label the vertices of $P_4$ by $v_1, \dots , v_4$ so that the edges are $v_1v_2$ , $v_2v_3$ , and $v_3v_4$ . Since $v_1v_2$ is an edge and $v_1v_3$ is not an edge, $v_1 + v_2> 1 > v_1 + v_3$ . In particular, $v_2> v_3$ . However, since $v_3v_4$ is an edge and $v_2v_4$ is not an edge, we have $v_2 < v_3$ , a contradiction.

Suppose that we remove an edge $bx$ from the torero graph where $b < x$ . Since this is normally an edge, we have that $b + x> 1$ . There is therefore some $y \in \mathbb {Q}$ such that $b < y < x$ and $b + y> 1$ . Now choose a and z such that $a + y < 1 < a + x $ and $z + b < 1 < z + y$ , which is possible since $b < y < x$ . This gives an induced copy of the bull when we remove a single edge from the torero graph. Since at each step we are choosing any rational in an interval, of which there are infinitely many, we can always choose a vertex such that no edge/non-edge to any of the preceding vertices has been perturbed. This means that we can also manage all locally finite edits. Adding an edge follows a similar argument where the new edge forms the edge $by$ in the bull. Diagrams of both cases can be seen in Figure 7.

Figure 7:

We show in (a) how to create a bull when the edge $bx$ is removed and we show in (b) how to create a bull when $by$ is added.

Finally, we explain how to handle any graph H whose (1,1)-core is a copy of the bull graph or $P_4$ . Since the torero graph has no induced copy of $P_4$ , it also has no copy of H. To embed a copy of H after a locally finite perturbation, we first embed a copy of the bull graph or $P_4$ as appropriate. The graph H may be obtained from either the bull or $P_4$ by iteratively adding vertices connected to none of the preceding vertices, or to all of them. Given any set of vertices, there are infinitely many vertices which are connected to every vertex in the set (those sufficiently close to 1) and infinitely many vertices which are connected to none of the vertices (those sufficiently close to 0), and we can find a copy of H by iteratively choosing such vertices.

7.3 Computer check for gatekeepers

Let us say that a 2-cut $\{u,v\}$ of G is an edge-2-cut if $\{u,v\}$ is an edge in G, and otherwise we say that it is a non-edge-2-cut.

Let H be a 2-connected graph and let $e=xy$ be an edge of H. We show that if there is no non-edge-2-cut $\{u,v\}$ of H such that a component of $H - \{u,v\}$ is an induced subgraph of $H-e$ , then e is a gatekeeper. Suppose toward a contradiction that the edge $e = xy$ is not a gatekeeper. Then there is some graph G which does not contain a copy of H such that gluing $H - e$ to G by identifying x and y with the endpoints of a non-edge of G creates a copy $H'$ of H. The vertices x and y are a $2$ -cut of $H'$ and removing them splits $H'$ into components, say, $C_1, \dots , C_r$ . At least one of these components is disjoint from G, else $H'$ would be contained entirely in G. Let u and v be the vertices from H that correspond to x and y in $H'$ . Then the component contained in $H' -\{x,y\}$ corresponds to a component of $H-\{u,v\}$ which is an induced subgraph of $H-e$ .

In fact, we can strengthen this condition slightly by including how a component of $H - \{u,v\}$ connects to the vertices $\{u,v\}$ , and this is what we use in practice. That is, for each non-edge-2-cut $\{u,v\}$ which splits H into components $C_1, \dots , C_r$ (say), we check if there is a copy of $H[V(C_i) \cup \{u,v\}]$ in $H-e$ where the vertices $\{u,v\}$ correspond to $\{x,y\}$ (either way round). An example is given in Figure 8.

Figure 8:

(a) The graph Dr[ with its only 2-cut highlighted in red. There are no 2-cuts which are edges, so every non-edge is a gatekeeper. (b) The graph with the edge $xy$ removed, with x and y highlighted in red. (c) The two fragments of the graph created by taking the components of the only 2-cut and adding back in the 2-cut. Observe that there is no copy of either of the fragments in (b) with matching vertex colors, which means that $xy$ is a gatekeeper.

For the constructed graph to be strongly-H-induced-saturated, it is not sufficient to simply glue on a copy of $H-e$ , even when e is a gatekeeper. Instead, we will replace each vertex in $H-e$ except for x and y by either infinite cliques or independent sets. As seen in the proof of Lemma 4.1, this can be done if neither vertex is a true twin or if neither vertex is a false twin, and we check for this condition before we check if a particular edge or non-edge is a gatekeeper using the above method.

Let us summarize by describing the implementation. First, the code checks that the graph is 2-connected and not a complete graph. Then all of the 2-cuts are enumerated and split into edge-2-cuts and non-edge-2-cuts. For each edge-2-cut $e=xy$ , we check that x and y do not have twins of differing types, and we move onto the next edge-2-cut if they do. We then replace e by a “red” edge, which serves to color the two vertices x and y. We loop over the non-edge-2-cuts and, for each non-edge-2-cut $\{u,v\}$ , we find the connected components of $H - \{u,v\}$ . For each connected component, C, we form the subgraph $H[V(C) \cup \{u,v\}]$ and add in a red edge between u and v. If this subgraph is isomorphic (including edge colors) to a subgraph of H, then we move onto another edge-2-cut. If we did not find such a subgraph for any choice of non-edge-2-cut and connected component, then we have found the suitable fixing operation and we move onto non-edge-2-cuts, which are handled similarly.

This gives us a method to look for fixing operations in a given $2$ -connected graph, but we can combine this with Lemma 4.6 to cover many more graphs. Indeed, suppose that the k-core of H, which we denote $H'$ , has a fixing operation (potentially found by the method above) for the class of $H'$ -free graphs. Then, by repeatedly applying Lemma 4.6, the graph H also admits a fixing operation for the class of $H'$ -free graphs and, in particular, is strongly saturating. Using the third and fourth parts of Lemma 4.6, we can also generalize this to what we call the 2-edge-core and the 2-non-edge-core. The 2-edge-core (resp. 2-non-edge-core) of a graph is formed by repeatedly removing vertices of degree less than $2$ and vertices of degree 2 whose neighbors are adjacent (resp. non-adjacent). The 2-edge-core of a graph has minimum degree at least 2 and no vertex v with degree $2$ whose neighbors are adjacent. Using Lemma 4.6 repeatedly, it follows that the graph H admits a fixing operation if its 2-edge-core or 2-non-edge-core admits a fixing operation, as required for check 7 above.

7.4 The rational geometric graph

The rational geometric graph is the graph on $\mathbb {Q}$ where there is an edge between q and r if and only if $|q - r| < \pi $ . Our main interest in this construction is to handle the three problematic small graphs shown in Figure 9. First, we make the easy observation that the neighborhood of any vertex v in the rational geometric graph can be partitioned into two cliques (where there may be edges between the cliques). In particular, this immediately shows that none of the graphs in Figure 9 are induced subgraphs of the rational geometric graph.

Figure 9:

The three problematic graphs which we handle using the rational geometric graph. The vertices highlighted in red have neighborhoods which cannot be split into two cliques.

To handle these graphs, what remains to show is that making any locally finite edit to the rational geometric graph creates an induced copy. Let us first consider making just a single change, either swapping an edge to a non-edge or a non-edge to an edge. By the symmetry of the rational geometric graph, we can assume that the edge/non-edge is from $0$ to some $r> 0$ , where $r < \pi $ if it is an edge and $r> \pi $ if it is a non-edge. It is straightforward to find a sequence of intervals from which we can choose the vertices of our small graphs, but in the interest of conciseness, we only sketch the constructions in Figure 10.

Figure 10:

The left-hand side shows the constructions when removing an edge on the dotted line and the right-hand side shows the constructions when adding the edge indicated by the dotted line.

Again, since we may choose from infinitely many vertices at any stage, we can always choose one such that none of the edges/non-edges to the proceeding vertices have been altered.

7.5 The final graph

There is one final graph we need to handle, shown in Figure 11.

Figure 11:

The final graph F?q~w.

We first give an auxiliary construction and then blow this up to allow for locally finite perturbations. Let G be the graph with vertex set $\mathbb {Z}^3$ . We join the vertex $(i,j, k)$ to $(a,b,c)$ in G if they agree in at least one coordinate, that is, $i = a$ , $j = b,$ or $k = c$ . There are two different kinds of edges $uv$ in G: those where u and v agree in exactly one coordinate and those where u and v agree in exactly two coordinates. There is one type of non-edge.

The graph $G'$ that we are interested in is obtained by blowing up each vertex of G into an infinite clique (replacing edges by complete bipartite graphs). We can split the neighbors of any vertex of $G'$ into three cliques based on which coordinate they agree on in G, which shows that $G'$ does not contain a copy of H.

Next, we show that making any locally finite perturbation creates a copy of H. We again first show that a single perturbation in G results in a copy of H in Figure 12.

Figure 12:

Examples of how to embed H into a copy of G when an edge of G has been perturbed. The dotted lines represent the location of the edge which was removed (top) or added (bottom).

Suppose now that $v_1'v_2'$ is perturbed in a locally finite perturbation of $G'$ where $v_1'$ and $v_2'$ correspond to different vertices $v_1$ and $v_2$ in G. Then a copy of H will still be created. The argument for this is similar to previous constructions: if $v_1,\dots ,v_k$ are the vertices in G on which a copy of H is created after perturbing $v_1v_2$ in G, then we may iteratively choose $v_i'$ corresponding to $v_i$ such that the edges/non-edges from $v_i'$ to $v_1',\dots ,v_{i-1}'$ have not been adjusted.

We have also introduced a new type of edge in $G'$ : an edge $u'u"$ where both $u'$ and $u"$ come from the infinite clique that corresponds to a single vertex u of G. We indicate in Figure 13 how to obtain a copy of H in a locally finite perturbation that only affects edges of this type.

Figure 13:

The construction shows how to obtain a copy of H when an edge is removed between two vertices which come from the same vertex of G (in this case, $(1,0,1)$ ).

7.6 Proof of Theorem 1.1

We now have all the components required to prove Theorem 1.1.