Bipartite completion of colored graphs avoiding chordless cycles of given lengths

• Published in: Discrete Applied Mathematics Vol. 318; pp. 97 - 112
• Main Authors: Eschen, Elaine M., Sritharan, R.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.09.2022
• Subjects: Bipartite completion; Chordal bipartite graphs; Graph sandwich problem; Graph sandwich problem; Bipartite completion; Chordal bipartite graphs
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2022.03.027

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 C6, C8, ⋯, C2p, 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+nα(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 (C6,C8,…,C2p)-free is NP-complete. Also, the corresponding graph sandwich problems are hard.