On the Tree–width of a Graph

Robertson and Seymour introduced the concept of the tree-width of a graph. It plays an important role in their results on graph minors culminating in their proof of Wagner’s conjecture. This concept seems to be interesting from the algorithmic point of view as well: many graph problems that are NP-complete in general can be polynomially solvable if graphs are constrained to have bounded tree-width [2]. In the present paper several equivalent definitions of tree-width are discussed and tree-width of several families of graphs is determined. All graphs in this paper are nonempty, finite and may have loops or multiple edges. V (G) and E(G) denote the set of vertices and the set of edges of the graph G, respectively. ω(G) denotes the cardinality of the maximal clique of G. Kk stands for the complete graph on k-vertices. For any subset X of V (G) we denote by G−X the graph obtained from G by deleting the vertices of X (and all the edges adjacent with them). For any edge e ∈ E(G) let G−e stand for the graph obtained from G by deleting e. In this paper the difference between isomorphism and equality of graphs is ignored. Definition 1. A tree-decomposition of a graph G is a pair (T, ), where T is a tree and = (Xt, t ∈ V (T )) is a family of subsets of V (G) with the following properties: (i) ∪(Xt, t ∈ V (T )) = V (G); (ii) for every edge e ∈ E(G) there exists t ∈ V (T ) such that e has both ends in Xt; (iii) for t, t′, t′′ ∈ V (T ), if t′ is on the path of T between t and t′′ then Xt ∩X ′′ t ⊆ X ′ t. The width of the tree-decomposition (T, ) is max t∈V (T ) (|Xt| − 1). Received May 22, 1992. 1980 Mathematics Subject Classification (1991 Revision). Primary 05C05, 05C75.