Connected and Loosely Connected List Homomorphisms

Abstract

In this paper, we study a special graph colouring problem, called the list homomorphism problem, which is a generalisation of the list colouring problem. Several variants of the list homomorphism problem have been considered before. In particular, a complete complexity classification of the connected list homomorphism problem for refiexive graphs has been given before, according to which the problem is polynomial time solvable for refiexive chordal graphs, and NP-complete for refiexive non-chordal graphs. A natural analogue of this result is known not to hold for this problem for bipartite graphs. We observe that the notion of list connectivity in the problem needs to be modified for bipartite graphs. We introduce a new variant called the bipartite loosely connected list homomorphism problem for bipartite graphs. We give a complete complexity classification of this problem, showing that it is polynomial time solvable for chordal bipartite graphs, and NP-complete for non-chordal bipartite graphs. This result is analogous to the result for the connected list homomorphism problem for refiexive graphs. We present a linear time algorithm for the bipartite loosely connected list homomorphism problem for chordal bipartite graphs, as well as for the connected list homomorphism problem for refiexive chordal graphs, showing that the algorithms can decide just by testing whether or not the corresponding consistency tests succeed.