Two Tree-Width-Like Graph Invariants

In this paper we introduce two tree-width-like graph invariants. The first graph invariant, which we denote by ν =(G), is defined in terms of positive semi-definite matrices and is similar to the graph invariant ν(G), introduced by Colin de Verdière in [J. Comb. Theory, Ser. B., 74:121–146, 1998]. The second graph invariant, which we denote by θ(G), is defined in terms of a certain connected subgraph property and is similar to λ(G), introduced by van der Holst, Laurent, and Schrijver in [J. Comb. Theory, Ser. B., 65:291–304, 1995]. We give some theorems on the behaviour of these invariants under certain transformations. We show that ν =(G)=θ(G) for any graph G with ν =(G)≤4, and we give minimal forbidden minor characterizations for the graphs satisfying ν =(G)≤k for k=1,2,3,4.

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Correspondence to Hein van der Holst.

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This paper is extracted from two chapters of [7]. This work was done while the author was at the Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.

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van der Holst, H. Two Tree-Width-Like Graph Invariants. Combinatorica 23, 633–651 (2003).

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Mathematics Subject Classification (2000):

  • 05C83
  • 05C50
  • 15A18