Bipartite completion of colored graphs avoiding chordless cycles of given lengths

Abstract

We consider a well-known restriction of the graph sandwich problem: Given a graph G with a proper vertex coloring, determine if there is a completion of G (formed by adding edges to G while maintaining the proper coloring) that has property P . We are interested in completions of G that are bipartite graphs without induced cycles of prescribed lengths. It is known that deciding whether there is a chordal bipartite completion, namely one without induced cycles on six or more vertices, is NP-complete when the input graph is colored with an arbitrary number of colors. We consider the case when the input graph is 3-colored and show that the problem of deciding whether there is a bipartite completion that avoids induced cycles C 6 , C 8 , ⋯ , C 2 p , for fixed p ≥ 3 , is NP-complete. In contrast, we characterize chordal bipartite completions of 3-colored graphs, and based on this, show that deciding whether a 3-colored graph can be completed to be chordal bipartite is solvable in O( m + n α ( n ) ) time. When the input admits a chordal bipartite completion, a size- n representation of a chordal bipartite completion can be constructed within the same time bound. It follows from our results that for every fixed k ≥ 3 , and for every fixed p ≥ 3 , deciding whether a k -colored graph can be completed to be bipartite and ( C 6 , C 8 , … , C 2 p ) -free is NP-complete. Also, the corresponding graph sandwich problems are hard.