Tree decompositions for a class of graphs

Abstract

For a graph G, if E( G) can be partitioned into several pairwise disjoint sets as { E 1, E 2,…, E 1} such that for any i with 1 ⩽ i ⩽ l, the subgraph induced by E 1 in G is a tree of order k i , then G is said to have a { k 1, k 2, … , k 1}-tree-decomposition. Ringel [3], and Ouyang and Liu [2] proved that every 2-connected maximal planar bipartite (mpb) graph of order n has a { n − 1, n − 1}-tree-decomposition and { n, n − 2}-tree-decomposition, respectively. Kampen [1] proved that every maximal planar (mp) graph of order n has a { n − 1, n − 1, n − 1}-tree-decomposition. In this paper, we consider the following class of graphs including mpb and mp graphs: A graph G is called a P k-graph, if if | G| ⩾ 3, | E( G)| = k(| G| −2) and | E( H)|⩽ k(| H| −2) for any subgraph H of G with | H| ⩾ 3. We prove that (i) for any P 2-graph of order n ⩾ 3, it has both a { n, n − 2}-tree-decomposition and a { n − 1, n − 1}-tree-decomposition, and moreover, these two kinds of tree-decompositions can be transformed to each other; (ii) for any P 3-graph of order n ⩾ 4, it has three kinds of tree-decompositions: { n, n, n − 3}-, { n, n − 1, n − 2}- and { n − 1, n − 1, n − 1}-tree-decomposition, and moreover, they can be transformed to each other. Since 2-connected mpb graphs are P 2-graphs and mp graphs are P 3-graphs, the results mentioned above from [1–3] are immediately implied by our results. Furthermore, all tree-decompositions above can be found in polynomial time.