On caterpillar factors in graphs

Abstract

A caterpillar is either a $K_2$ or a tree on at least 3 vertices such that deleting its leaves we obtain a path of order at least 1. Given a simple undirected graph $G=(V,E)$, a caterpillar factor of G is a set of caterpillar subgraphs of G such that each vertex $v\in V$ belongs to exactly one of them. A caterpillar factor F is internally even if every vertex of degree ${\mathrm{deg}}_F(v)\ge 2$ has an even degree; F is odd if ${\mathrm{deg}}_F(v)$ is odd for every $v\in V(G)$. We present a linear-time algorithm that decides whether a tree admits an internally even caterpillar factor and, on the other hand, we prove that the decision problem is NP-complete on the class of planar bipartite graphs. For the odd caterpillar factor problem, we obtain similar results. It can be decided in linear time over the class of trees, but the problem is NP-complete on the class of bipartite graphs.