TY - GEN
T1 - On the Complexity of Minimum Maximal Uniquely Restricted Matching
AU - Chaudhary, Juhi
AU - Panda, B. S.
N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
PY - 2020/1/1
Y1 - 2020/1/1
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. A matching M in a graph G is called a uniquely restricted matching if G[V(M)], the subgraph of G induced by the M-saturated vertices of G, contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any other uniquely restricted matching of G. Given a graph G, the Min-Max-UR Matching problem asks to find a maximal uniquely restricted matching of minimum cardinality in G. In general, the decision version of the Min-Max-UR Matching problem is known to be NP-complete for general graphs and remains so even for bipartite graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete for chordal bipartite graphs and chordal graphs. On the positive side, we prove that the Min-Max-UR Matching problem is polynomial time solvable for bipartite permutation graphs and proper interval graphs. Finally, we show that the Min-Max-UR Matching problem is APX-complete for bounded degree graphs.
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. A matching M in a graph G is called a uniquely restricted matching if G[V(M)], the subgraph of G induced by the M-saturated vertices of G, contains exactly one perfect matching. A uniquely restricted matching M is maximal if M is not properly contained in any other uniquely restricted matching of G. Given a graph G, the Min-Max-UR Matching problem asks to find a maximal uniquely restricted matching of minimum cardinality in G. In general, the decision version of the Min-Max-UR Matching problem is known to be NP-complete for general graphs and remains so even for bipartite graphs. In this paper, we strengthen this result by proving that this problem remains NP-complete for chordal bipartite graphs and chordal graphs. On the positive side, we prove that the Min-Max-UR Matching problem is polynomial time solvable for bipartite permutation graphs and proper interval graphs. Finally, we show that the Min-Max-UR Matching problem is APX-complete for bounded degree graphs.
KW - APX-completeness
KW - Graph algorithms
KW - Matching
KW - Minimum maximal matching
KW - NP-completeness
KW - Uniquely restricted matching
UR - http://www.scopus.com/inward/record.url?scp=85097827801&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-64843-5_25
DO - 10.1007/978-3-030-64843-5_25
M3 - Conference contribution
AN - SCOPUS:85097827801
SN - 9783030648428
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 364
EP - 376
BT - Combinatorial Optimization and Applications - 14th International Conference, COCOA 2020, Proceedings
A2 - Wu, Weili
A2 - Zhang, Zhongnan
PB - Springer Science and Business Media Deutschland GmbH
T2 - 14th International Conference on Combinatorial Optimization and Applications, COCOA 2020
Y2 - 11 December 2020 through 13 December 2020
ER -