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.