Decompositions of critical trees with cutwidth k

Abstract

The cutwidth of a graph G is the minimum integer k such that the vertices of G are arranged in a linear layout $[v_1,\ldots ,v_n]$ in such a way that, for every $j= 1,\ldots ,n-1$, there are at most k edges with one endpoint in $\{v_1,\ldots ,v_j\}$ and the other in $\{v_{j+1},\ldots ,v_n\}$. The cutwidth problem for G is to determine the cutwidth k of G. A graph G with cutwidth k is k-cutwidth critical if every proper subgraph of G has cutwidth less than k and G is homeomorphically minimal. In this paper, we obtain that any k-cutwidth critical tree $\mathcal {T}$ can be decomposed into three $(k-1)$-cutwidth critical subtrees for $k\ge 2$; And an $O(|V(\mathcal {T})|^{2}\mathrm{log} |V(\mathcal {T})|)$ algorithm of computing the three subtrees is given.