On Strong Tree-Breadth

Abstract

In this paper, we introduce and investigate a new notion of strong tree-breadth. We say that a graph G has strong tree-breadth $\rho$ if there is a tree-decomposition T for G such that each bag B of T is equal to the complete $\rho$-neighbourhood of some vertex v in G, i. e., $B = N_G^\rho [v]$. We show that

• it is NP-complete to determine if a given graph has strong tree-breadth $\rho$, even for $\rho = 1$;

• if a graph G has strong tree-breadth $\rho$, then we can find a tree-decomposition for G with tree-breadth $\rho$ in $\mathcal {O}(n^2m)$ time;

• with some additional restrictions, a tree-decomposition with strong breadth $\rho$ can be found in polynomial time;

• some graph classes including distance-hereditary graphs have strong tree-breadth 1.