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• W2949644883 abstract "The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollob'as and Sorkin as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors, 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, we prove that the one-variable vertex-nullity interlace polynomial is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollob'as and Riordan. We define the gamma invariant as the coefficient of x^1 in the vertex-nullity interlace polynomial, analogously to the beta invariant, which is the coefficient of x^1 in the Tutte polynomial. We then turn to distance hereditary graphs, and show that graphs in this class have gamma invariant of 2^{n+1} when n true twins are added in their construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with gamma invariant 2, just as the series-parallel graphs are exactly the class of graphs with beta invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of any Euler circuit in 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 distance hereditry graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs." @default.
• W2949644883 created "2019-06-27" @default.
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• W2949644883 date "2006-04-04" @default.
• W2949644883 modified "2023-09-27" @default.
• W2949644883 title "Distance Hereditary Graphs and the Interlace Polynomial" @default.
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• W2949644883 hasPublicationYear "2006" @default.
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