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.