TY - JOUR

T1 - Dominating induced matching in some subclasses of bipartite graphs

AU - Panda, B. S.

AU - Chaudhary, Juhi

N1 - Funding Information:
The authors would like to thank the anonymous referees for their helpful comments leading to improvements in the presentation of the paper. The first author wants to thank the SERB , Department of Science and Technology for their support vide Diary No. SERB/F/12949/2018-2019 . The second author wants to thank the Department of Science and Technology (INSPIRE), grant no: IF160665 , for their support.
Funding Information:
The authors would like to thank the anonymous referees for their helpful comments leading to improvements in the presentation of the paper. The first author wants to thank the SERB, Department of Science and Technology for their support vide Diary No. SERB/F/12949/2018-2019. The second author wants to thank the Department of Science and Technology (INSPIRE), grant no: IF160665, for their support.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/9/11

Y1 - 2021/9/11

N2 - A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. An edge e∈E is said to dominate itself and all other edges adjacent to it. A matching M in a graph G=(V,E) is called a dominating induced matching (d.i.m.) if every edge of G is dominated by edges of M exactly once. The dominating induced matching decide (DIM-DECIDE) problem asks whether a graph G contains a dominating induced matching. The dominating induced matching (DIM) problem asks to compute a dominating induced matching (d.i.m.) in a graph G that admits a dominating induced matching. The DIM-DECIDE problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of the DIM-DECIDE problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we characterize the class of star-convex bipartite graphs admitting a d.i.m. This characterization leads to a linear time algorithm to test whether a star-convex bipartite graph admits a d.i.m. and, if so, constructs a d.i.m. in such a star-convex bipartite graph in linear time. We also propose polynomial time algorithms to construct a d.i.m. in long-k-star-convex bipartite graphs as well as in circular-convex bipartite graphs if the input graph admits a d.i.m.

AB - A subset M⊆E of edges of a graph G=(V,E) is called a matching if no two edges of M share a common vertex. An edge e∈E is said to dominate itself and all other edges adjacent to it. A matching M in a graph G=(V,E) is called a dominating induced matching (d.i.m.) if every edge of G is dominated by edges of M exactly once. The dominating induced matching decide (DIM-DECIDE) problem asks whether a graph G contains a dominating induced matching. The dominating induced matching (DIM) problem asks to compute a dominating induced matching (d.i.m.) in a graph G that admits a dominating induced matching. The DIM-DECIDE problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen the NP-completeness result of the DIM-DECIDE problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs, a proper subclass of bipartite graphs. On the positive side, we characterize the class of star-convex bipartite graphs admitting a d.i.m. This characterization leads to a linear time algorithm to test whether a star-convex bipartite graph admits a d.i.m. and, if so, constructs a d.i.m. in such a star-convex bipartite graph in linear time. We also propose polynomial time algorithms to construct a d.i.m. in long-k-star-convex bipartite graphs as well as in circular-convex bipartite graphs if the input graph admits a d.i.m.

KW - Bipartite graphs

KW - Dominating induced matching

KW - NP-completeness

KW - Perfect elimination bipartite graphs

KW - Polynomial time algorithms

UR - http://www.scopus.com/inward/record.url?scp=85110097827&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2021.06.031

DO - 10.1016/j.tcs.2021.06.031

M3 - Article

AN - SCOPUS:85110097827

SN - 0304-3975

VL - 885

SP - 104

EP - 115

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -