Skip to main content

Juhi Chaudhary

Given a graph \(G=(V,E)\), a set \(M\subseteq E\) is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G[M], the subgraph of G induced by M, is same as G[S], the subgraph... more
Given a graph \(G=(V,E)\), a set \(M\subseteq E\) is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G[M], the subgraph of G induced by M, is same as G[S], the subgraph of G induced by \(S=\{v\in V |\) v is incident on an edge of \(M \}\). An induced matching M in a graph G is dominating if every edge not in M shares exactly one of its endpoints with a matched edge. The dominating induced matching (DIM) problem (also known as Efficient Edge Domination) is a decision problem that asks whether a graph G contains a dominating induced matching or not. This problem is NP-complete for general graphs as well as for bipartite graphs. In this paper, we show that the DIM problem is NP-complete for perfect elimination bipartite graphs and propose polynomial time algorithms for star-convex, triad-convex and circular-convex bipartite graphs which are subclasses of bipartite graphs.
A subset $$M\subseteq E$$ M ⊆ E of edges of a graph $$G=(V,E)$$ G = ( V , E ) is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G [ M ], the subgraph of G induced by M... more
A subset $$M\subseteq E$$ M ⊆ E of edges of a graph $$G=(V,E)$$ G = ( V , E ) is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G [ M ], the subgraph of G induced by M , is the same as G [ S ], the subgraph of G induced by $$S=\{v \in V |$$ S = { v ∈ V | v is incident on an edge of $$M\}$$ M } . The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. Given a graph G and a positive integer k , the Induced Matching Decision problem is to decide whether G has an induced matching of cardinality at least k . The Maximum Weight Induced Matching problem in a weighted graph $$G=(V,E)$$ G = ( V , E ) in which the weight of each edge is a positive real number, is to find an induced matching such that the sum of the weights of its edges is maximum. It is known that the Induced Matching Decision problem and hence the Maximum Weight Induced Matching problem is known to be NP-complete for general graphs and bipartite graphs. In this paper, we strengthened this result by showing that the Induced Matching Decision problem is NP-complete for star-convex bipartite graphs, comb-convex bipartite graphs, and perfect elimination bipartite graphs, the subclasses of the class of bipartite graphs. On the positive side, we propose polynomial time algorithms for the Maximum Weight Induced Matching problem for circular-convex bipartite graphs and triad-convex bipartite graphs by making polynomial time reductions from the Maximum Weight Induced Matching problem in these graph classes to the Maximum Weight Induced Matching problem in convex bipartite graphs.