1. Introduction
Weak saturation. Let
$F$
and
$H$
be graphs and let
$G$
be a spanning subgraph of
$F$
. We say that
$G$
is weakly
$H$
-saturated in
$F$
, if the edges of
$E(F) \setminus E(G)$
can be ordered into a sequence
$e_1, \dots , e_m$
in such a way that for every
$i \in [m]$
the graph obtained from
$G$
by adding the edges
$e_1, \dots , e_i$
contains a copy of
$H$
which contains
$e_i$
. The weak saturation number
$\mathrm{wsat}(F,H)$
is defined as the minimum number of edges of a graph which is weakly
$H$
-saturated in
$F$
.
The concept of weak saturation was first introduced in 1968 by Bollobás [Reference Bollobás4] who considered the case when
$F$
and
$H$
are complete graphs and conjectured that
$\mathrm{wsat}(K_n,K_t)=\binom {n}{2}-\binom {n-t+2}{2}$
. This was confirmed by Frankl [Reference Frankl10] and, independently, Kalai [Reference Kalai14, Reference Kalai15] (a version for matroids was proven earlier by Lovász [Reference Lovász16]) and extended by Alon [Reference Alon1] and Blokhuis [Reference Blokhuis3]. Subsequently, weak saturation was studied for different classes of graphs and (in the analogous setting) hypergraphs; see [Reference Alon1, Reference Balogh, Bollobás, Morris and Riordan2, Reference Balogh and Pete5, Reference Bulavka, Tancer and Tyomkyn6, Reference Erdős, Füredi and Tuza8, Reference Faudree and Gould9, Reference Morrison and Noel17, Reference Moshkovitz and Shapira18, Reference Pikhurko20, Reference Pikhurko21, Reference Shapira and Tyomkyn23, Reference Terekhov and Zhukovskii27, Reference Terekhov and Zhukovskii28, Reference Tuza25, Reference Tuza26]. As a common theme, upper bounds on
$wsat(F,H)$
are usually established via simple constructions, while proving lower bounds tends to be much harder and typically requires methods from algebra or geometry.
In this note, we show that any kind of classification of weak saturation numbers in full generality is hopeless unless P = NP. More concretely, we show that determining the weak saturation number is already hard in a seemingly very simple case when
$H$
is the complete graph on three vertices
$K_3$
:
Theorem 1.
Given a graph
$F$
with
$n$
vertices as input, it is NP-hard to decide whether
$\mathrm{wsat}(F, K_3) = n-1$
.
Note that
$\mathrm{wsat}(F, K_3) \geq n-1$
for any connected
$n$
-vertex graph
$F$
, as any weakly
$K_3$
-saturated graph in
$F$
must be spanning. In our reduction,
$F$
will always be connected.
Our main tool to prove Theorem 1 is to establish a connection between weak saturation and shellability (and collapsibility) of simplicial complexes. We point out that a recent preprint [Reference Chakraborti, Cho, Kim and Kim7] establishes a connection between weak saturation and
$d$
-collapsibility (closely related to collapsibility) in the context of fractional Helly-type theorems. Our setting, however, is quite different.
Simplicial complexes, shellability, and collapsibility. Let us recall that a(n abstract) simplicial complex is a set system
$K$
such that if
$\sigma \in K$
and
$\tau \subseteq \sigma$
, then
$\tau \in K$
. We will consider only finite simplicial complexes. The set of vertices of
$K$
is the set
$\bigcup K$
. The elements of
$K$
are called faces of
$K$
and the dimension of a face
$\sigma \in K$
is defined as
$\dim \sigma\, {:}{=}\ |\sigma | - 1$
. The faces of dimension
$1$
are edges and the faces of dimension
$2$
are triangles. (A triangle in a simplicial complex should not be confused with a graph-theoretic triangle, a copy of
$K_3$
. We will avoid using the notion “triangle” in the latter context.) The dimension of the complex,
$\dim K$
is defined as the maximum of the dimensions of faces in
$K$
. Given a non-negative integer
$k$
, the
$k$
-skeleton of a simplicial complex
$K$
is a subcomplex of
$K$
denoted
$K^{(k)}$
consisting of faces of
$K$
of dimension at most
$k$
. From now on we regard graphs as (at most)
$1$
-dimensional simplicial complexes. In particular, the
$1$
-skeleton of a simplicial complex is a graph.
Given a sequence
$\vartheta _1, \dots , \vartheta _k$
of faces of
$K$
we denote by
$K[\vartheta _1, \dots , \vartheta _k]$
the subcomplex of
$K$
induced by these faces, that is, the subcomplex formed by faces
$\sigma$
such that
$\sigma \subseteq \vartheta _i$
for some
$i \in [k]$
. An inclusion maximal face of a simplicial complex is a facet and a simplicial complex is pure if all facets have the same dimension. A pure
$d$
-dimensional complex
$K$
is shellable if there is an ordering
$\vartheta _1, \dots , \vartheta _m$
of all facets of
$K$
such that for every
$i \in \{2, \dots , m\}$
the complex
$K[\vartheta _i] \cap K[\vartheta _1, \dots , \vartheta _{i-1}]$
is pure and
$(d-1)$
-dimensional.Footnote
1
A simplicial complex
$K'$
arises from
$K$
by an elementary collapse if there is a face
$\tau$
of
$K$
contained in a single facet
$\sigma$
distinct from
$\tau$
and
$K'$
is obtained from
$K$
by removing all faces containing
$\tau$
. A simplicial complex
$K$
collapses to a subcomplex
$L$
, if there is a sequence
$K = K_1, K_2, \dots , K_\ell = L$
of simplicial complexes such that
$K_{i+1}$
is obtained from
$K_i$
by an elementary collapse for
$i \in [\ell - 1]$
. A simplicial complex
$K$
is collapsible if it collapses to a point (an arbitrary vertex of
$K$
).
The reduced Euler characteristic of a complex
$K$
is defined as
\begin{equation*} \tilde \chi (K) = \sum _{i=-1}^{\dim K} (-1)^i f_i(K) \end{equation*}
where
$f_i(K)$
is the number of
$i$
-dimensional faces of
$K$
. (Note that the empty set has the dimension equal to
$-1$
.) Given a simplicial complex
$K$
, its barycentric subdivision
$\mathrm{sd} K$
is a complex whose vertices are nonempty faces of
$K$
and whose faces are collections
$\{\vartheta _1, \dots , \vartheta _k\}$
of faces of
$K$
with
$\emptyset \neq \vartheta _1 \subsetneq \vartheta _2 \subsetneq \cdots \subsetneq \vartheta _k$
.
Hardness of shellability. In [Reference Goaoc, Paták, Patáková, Tancer and Wagner11], Goaoc, Paták, Patáková, Tancer, and Wagner proved that shellability is NP-hard. We state a corollary of the main technical proposition from [Reference Goaoc, Paták, Patáková, Tancer and Wagner11] in a way convenient for us. In the statement, we use 3-CNF formulas, that is, formulas in a conjunctive normal form where each clause contains three literals. We skip details, referring the reader to [Reference Goaoc, Paták, Patáková, Tancer and Wagner11], as we use 3-CNF formulas only implicitly. We only need the fact that the decision problem of whether a 3-CNF formula is satisfiable is a well-known NP-hard problem, known as 3-satisfiability.
Theorem 2 (Essentially Proposition 8 from [Reference Goaoc, Paták, Patáková, Tancer and Wagner11]). There is a polynomial time algorithm that produces from a given 3-CNF formula
$\phi$
with
$t$
variables a pure connected 2-dimensional complex
$K_\phi$
with
$\tilde \chi (K_\phi ) = t$
such that the following statements are equivalent:
-
(i) The formula
$\phi$
is satisfiable.
-
(ii) The second barycentric subdivision
$\mathrm{sd}^2 K_\phi$
is shellable.
-
(ii’) The third barycentric subdivision
$\mathrm{sd}^3 K_\phi$
is shellable.
-
(ii’’) The forth barycentric subdivision
$\mathrm{sd}^4 K_\phi$
is shellable.
-
(iii) The complex
$K_\phi$
is collapsible after removing some
$t$
triangles.
-
(iii’) The barycentric subdivision
$\mathrm{sd} K_\phi$
is collapsible after removing some
$t$
triangles.
-
(iii’) The second barycentric subdivision
$\mathrm{sd}^2 K_\phi$
is collapsible after removing some
$t$
triangles.
We are really interested only in the items (i), (ii), and (iii”). The remaining items are auxiliary for explaining the proof.
Because Proposition 8 from [Reference Goaoc, Paták, Patáková, Tancer and Wagner11] is not formulated exactly this way, we briefly explain how Theorem 2 follows from [Reference Goaoc, Paták, Patáková, Tancer and Wagner11]: The construction of
$K_\phi$
is according to [Reference Goaoc, Paták, Patáková, Tancer and Wagner11, Proposition 8]. The fact that the number of variables of
$\phi$
equals
$\tilde \chi (K_\phi )$
is the content of [Reference Goaoc, Paták, Patáková, Tancer and Wagner11, Proposition 12]. Then the statements (i), (ii), (ii’), (iii), and (iii’) of Theorem 2 are explicitly stated as equivalent statements in the (joint) proof of Theorems 4 and 5 in [Reference Goaoc, Paták, Patáková, Tancer and Wagner11]. It remains to argue that (ii”) and (iii”) are equivalent as well. The proof of Theorems 4 and 5 in [Reference Goaoc, Paták, Patáková, Tancer and Wagner11] contains in particular implications (ii)
$\Rightarrow$
(ii’)
$\Rightarrow$
(iii’)
$\Rightarrow$
(i). The implications (ii’)
$\Rightarrow$
(ii”)
$\Rightarrow$
(iii”)
$\Rightarrow$
(i) work with the exactly same reasoning, which proves the equivalence.Footnote
2
Finally, it is possible to check that
$K_\phi$
is connected directly from the construction in [Reference Goaoc, Paták, Patáková, Tancer and Wagner11]. Alternatively, Skotnica and Tancer proved [Reference Skotnica and Tancer24, Appendix A] that
$K_\phi$
from exactly this construction is homotopy equivalent to the wedge of spheres. This also implies that
$K_\phi$
is connected.
In our proof of Theorem 1, we use Theorem 2 with
$L_\phi = \mathrm{sd}^2 K_\phi$
, extending it to the following setting.
Theorem 3.
There is a polynomial time algorithm that produces from a given 3-CNF formula
$\phi$
with
$t$
variables a pure 2-dimensional connected complex
$L_\phi$
with
$\tilde \chi (L_\phi ) = t$
such that the following statements are equivalent:
-
(i) The formula
$\phi$
is satisfiable.
-
(ii) The complex
$L_\phi$
is shellable.
-
(iii) The complex
$L_\phi$
is collapsible after removing some
$t$
triangles.
-
(iv) We have
$\mathrm{wsat}(L_\phi ^{(1)}, K_3) = n-1$
where
$n$
is the number of vertices of
$L_\phi$
. -
(v) The complex
$L_\phi$
is collapsible after removing some number of triangles.
The proof of Theorem 3 is given in the next section. Theorem 1 follows immediately from the equivalence of (i) and (iv) in Theorem 3.
Proof of Theorem 1.
The equivalence of (i) and (iv) in Theorem 3 provides a polynomial time reduction from 3-satisfiability to determining whether
$\mathrm{wsat}(F, K_3) = n-1$
. (Note that the graph
$L_\phi ^{(1)}$
can be constructed from
$L_\phi$
in polynomial time.) Given that 3-satisfiability is NP-hard, it follows that the latter problem is NP-hard as well.
Finally, we remark that the items (ii), (iii) and (v) in Theorem 3 are again only auxiliary in order to show conveniently the equivalence of (i) and (iv).
2. The proof of Theorem 3
The aim of this section is to prove Theorem 3, completing the proof of Theorem 1.
As stated earlier, given a 3-CNF formula
$\phi$
we take
$K_\phi$
from Theorem 2 and set
$L_\phi \, {:}{=}\ \mathrm{sd}^2 K_\phi$
. Given that
$K_\phi$
is
$2$
-dimensional, the complexity of
$L_\phi$
grows only by a constant factor when compared with
$K_\phi$
, and so
$L_\phi$
can be constructed in polynomial time in the size of
$K_\phi$
, hence in polynomial time in the size of
$\phi$
. The complex
$L_\phi$
is connected because
$K_\phi$
is connected. We also remark that
$\tilde \chi (L_\phi ) = \tilde \chi (K_\phi ) = t$
because a complex and its barycentric subdivision have the same reduced Euler characteristic.Footnote
3
The items (i), (ii), and (iii) of Theorem 3 are equivalent due to Theorem 2. We will now show implications (ii)
$\Rightarrow$
(iv)
$\Rightarrow$
(v)
$\Rightarrow$
(iii), completing the proof.
Proof of (ii) ⇒ (iv).
Consider a shelling
$\vartheta _1, \dots , \vartheta _m$
of
$L_\phi$
, that is, a sequence of all facets of
$L_\phi$
witnessing that
$L_\phi$
is shellable. Given that
$L_\phi$
is pure 2-dimensional, all facets are triangles.
We construct a spanning tree
$G$
in the 1-skeleton
$L_\phi ^{(1)}$
in the following way. First we set
$G_1$
inside
$\vartheta _1$
to contain all three vertices and two arbitrarily chosen edges. Next, for
$i \in \{2, \dots , m\}$
, we inductively assume that we have a spanning tree
$G_{i-1}$
of
$L_\phi [\vartheta _1, \dots , \vartheta _{i-1}]^{(1)}$
and we construct a spanning tree
$G_i$
of
$L_\phi [\vartheta _1, \dots , \vartheta _{i}]^{(1)}$
; see Fig. 1 for an illustration. We distinguish two cases. If
$\vartheta _i$
meets the preceding triangles in two or three edges we set
$G_i\, {:}{=}\ G_{i-1}$
. Note that
$L_\phi [\vartheta _1, \dots , \vartheta _{i-1}]$
and
$L_\phi [\vartheta _1, \dots , \vartheta _{i}]$
have the same sets of vertices in this case. Thus
$G_i$
is indeed a spanning tree. If
$\vartheta _i$
meets the preceding triangles in a single edge, then we add to
$G_{i-1}$
the new vertex of
$\vartheta _i$
(i.e., the vertex not contained in
$L_\phi [\vartheta _1, \dots , \vartheta _{i-1}]$
) and one edge inside
$\vartheta _i$
containing this vertex. Other cases are not possible because
$\vartheta _1, \dots , \vartheta _m$
is a shelling. We set
$G \, {:}{=}\ G_m$
.
Figure 1. The figure displays three options of how
$\vartheta _i$
may meet
$L_\phi [\vartheta _1,\dots ,\vartheta _{i-1}]$
according to the number of shared edges. (The third displayed case is not fully realistic globally because the displayed
$L_\phi [\vartheta _1,\dots ,\vartheta _{i-1}]$
is not shellable. However, it becomes realistic if we assume that the outer face is also part of the complex and it is actually
$\vartheta _1$
.)
In order to finish the proof, we claim that
$G$
is weakly
$K_3$
-saturated in
$L_\phi ^{(1)}$
. Indeed, the saturating sequence follows the shelling. The first edge
$e_1$
is the unique edge of
$\vartheta _1$
not contained in
$G_1$
. This edge completes a copy of
$K_3$
inside
$\vartheta _1$
(more precisely
$L_\phi [\vartheta _1]^{(1)}$
). Next, for
$i \in \{2, \dots , m\}$
, we observe that
$L_\phi [\vartheta _1, \dots , \vartheta _{i}]^{(1)}$
contains at most one more edge not contained in
$G_i$
than
$L_\phi [\vartheta _1, \dots , \vartheta _{i-1}]^{(1)}$
. It is exactly one edge
$e_i$
in the cases that
$\vartheta _i$
meets the preceding triangles in one or two edges; see again Fig. 1 for an example. (With a slight abuse of notation we denote the edge by
$e_i$
though it may be not the
$i$
-th edge in the order, if some preceding edges are missing.) The edge
$e_i$
again completes the copy of
$K_3$
inside
$\vartheta _i$
, thereby inside
$L_\phi [\vartheta _1, \dots , \vartheta _{i}]^{(1)}$
.
Proof of (iv) ⇒ (v).
The facts that
$L_{\phi }$
is connected and
$\mathrm{wsat}(L_\phi ^{(1)}, K_3) = n-1$
imply that there exists a spanning tree
$G$
weakly
$K_3$
-saturated in
$L_\phi ^{(1)}$
. We will show that
$L_\phi ^{(1)}$
collapses to
$G$
. This implies that
$L_\phi ^{(1)}$
is collapsible as any tree is collapsible.
Let
$e_1, \dots , e_m$
be a sequence of edges witnessing that
$G$
is weakly
$K_3$
-saturated in
$L_\phi ^{(1)}$
. For every such edge
$e_i$
we fix a copy
$J_i$
of
$K_3$
it creates. Now we crucially use that
$L_\phi$
is a barycentric subdivision of another complex. It is well known and not hard to show (at least for
$2$
-complexes) that every copy of
$K_3$
induces a triangle in
$L_\phi$
. (In general barycentric subdivisions are flag, that is, every clique in the 1-skeleton induces a full simplex in the complex.) By
$\vartheta _i$
we denote the triangle induced by
$J_i$
. We remark that the triangles
$\vartheta _i$
are distinct because for
$i \lt j$
,
$\vartheta _j$
contains the edge
$e_j$
while
$\vartheta _i$
does not contain it.
We set
$L$
to be
$L_\phi$
after removing all triangles that do not appear as
$\vartheta _i$
for some
$i \in [m]$
. Now we perform elementary collapses on
$L$
in the reverse order of
$e_1, \dots , e_m$
. That is, we first claim that
$e_m$
is in a unique triangle
$\vartheta _m$
. This is indeed the case as the triangles
$\vartheta _i$
with
$i \lt m$
do not contain
$e_m$
. We perform an elementary collapse on
$L$
removing
$e_m$
and
$\vartheta _m$
. After performing this collapse we claim that
$e_{m-1}$
is in a unique triangle
$\vartheta _{m-1}$
. This is indeed the case as
$\vartheta _m$
has been already removed and the triangles
$\vartheta _i$
with
$i \lt m - 1$
do not contain
$e_{m-1}$
. We perform an elementary collapse on the intermediate complex removing
$e_{m-1}$
and
$\vartheta _{m-1}$
. We continue this until we have collapsed
$L$
to
$G$
as required. (Note that every edge of
$L$
outside
$G$
appears as some
$e_i$
because
$e_1, \dots , e_m$
witnesses that
$G$
is weakly
$K_3$
saturated in
$L_\phi ^{(1)}$
and
$L_\phi ^{(1)} = L^{(1)}$
.)
Proof of (v) ⇒ (iii).
Let
$L$
be a collapsible complex obtained from
$L_\phi$
by removing
$k$
triangles. We know that
$\tilde \chi (L_\phi ) = t$
, so
$\tilde \chi (L) = t - k$
follows immediately from the definition of the reduced Euler characteristic. Because collapses preserve the homotopy type (this is explained, e.g., in a slightly more general setting in [Reference Rourke and Sanderson22, Chapter 3]) and the (reduced) Euler characteristic is an invariant of the homotopy type (see [Reference Hatcher13, Theorem 2.44]), we deduce
$\tilde \chi (L) = \tilde \chi (pt)$
where
$pt$
stands for a point.Footnote
4
However,
$\tilde \chi (pt) = 0$
(the empty set and the point itself contribute
$-1$
and
$1$
, respectively). Thus we deduce
$k = t$
, which proves (iii).
Acknowledgements
We would like to thank Adam Rajský for helpful early discussions.
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