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Distance Hereditary Graphs and the Interlace Polynomial

Published online by Cambridge University Press:  01 November 2007

JOANNA A. ELLIS-MONAGHAN  and
IRASEMA SARMIENTO
JOANNA A. ELLIS-MONAGHAN
Affiliation:
Department of Mathematics, Saint Michael's College, 1 Winooski Park, Colchester, VT 05439, USA (e-mail: jellis-monaghan@smcvt.edu
IRASEMA SARMIENTO
Affiliation:
Dipartimento di Matematica, Universitàdi Roma ‘Tor Vergata’, Via della Ricerca Scientifica, I-00133, Rome, Italy (e-mail: sarmient@mat.uniroma2.it
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Abstract

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].

We define the γ invariant as the coefficient of x1 in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1 in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1 when n true twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

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Paper
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Combinatorics, Probability and Computing , Volume 16 , Issue 6 , November 2007 , pp. 947 - 973
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Copyright © Cambridge University Press 2007

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