TY - JOUR

T1 - Acyclic matching in some subclasses of 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 thanks 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 thanks 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:
© 2022 Elsevier B.V.

PY - 2023/1/17

Y1 - 2023/1/17

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. Given a matching M in G, a vertex v∈V is called M-saturated if there exists an edge e∈M incident with v. A matching M of a graph G is called an acyclic matching if, G[V(M)], the subgraph of G induced by the M-saturated vertices of G is an acyclic graph. Given a graph G, the ACYCLIC MATCHING problem asks to find an acyclic matching of maximum cardinality in G. The DECIDE-ACYCLIC MATCHING problem takes a graph G and an integer k and asks whether G has an acyclic matching of cardinality at least k. The DECIDE-ACYCLIC MATCHING problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen this result by showing that the DECIDE-ACYCLIC MATCHING problem remains NP-complete for comb-convex bipartite graphs, star-convex bipartite graphs, and dually chordal graphs. On the positive side, we show that the ACYCLIC MATCHING problem is linear time solvable for split graphs, block graphs, and proper interval graphs. We show that the ACYCLIC MATCHING problem is hard to approximate within a factor of n1−ϵ for any ϵ>0 unless P=NP. Also, we show that the ACYCLIC MATCHING problem is APX-complete for (2k+1)-regular graphs for every fixed integer k≥3.

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. Given a matching M in G, a vertex v∈V is called M-saturated if there exists an edge e∈M incident with v. A matching M of a graph G is called an acyclic matching if, G[V(M)], the subgraph of G induced by the M-saturated vertices of G is an acyclic graph. Given a graph G, the ACYCLIC MATCHING problem asks to find an acyclic matching of maximum cardinality in G. The DECIDE-ACYCLIC MATCHING problem takes a graph G and an integer k and asks whether G has an acyclic matching of cardinality at least k. The DECIDE-ACYCLIC MATCHING problem is known to be NP-complete for general graphs as well as for bipartite graphs. In this paper, we strengthen this result by showing that the DECIDE-ACYCLIC MATCHING problem remains NP-complete for comb-convex bipartite graphs, star-convex bipartite graphs, and dually chordal graphs. On the positive side, we show that the ACYCLIC MATCHING problem is linear time solvable for split graphs, block graphs, and proper interval graphs. We show that the ACYCLIC MATCHING problem is hard to approximate within a factor of n1−ϵ for any ϵ>0 unless P=NP. Also, we show that the ACYCLIC MATCHING problem is APX-complete for (2k+1)-regular graphs for every fixed integer k≥3.

KW - Acyclic matching

KW - APX-completeness

KW - Graph algorithms

KW - Matching

KW - NP-completeness

KW - Polynomial-time algorithms

UR - http://www.scopus.com/inward/record.url?scp=85145231284&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2022.12.008

DO - 10.1016/j.tcs.2022.12.008

M3 - Article

AN - SCOPUS:85145231284

SN - 0304-3975

VL - 943

SP - 36

EP - 49

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -