Results on vertex-edge and independent vertex-edge domination

Subhabrata Paul and Keshav Ranjan
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Subhabrata Paul: IIT Patna
Keshav Ranjan: IIT Madras

Journal of Combinatorial Optimization, 2022, vol. 44, issue 1, No 15, 303-330

Abstract: Abstract Given a graph $$G = (V,E)$$ G = ( V , E ) , a vertex $$u \in V$$ u ∈ V ve-dominates all edges incident to any vertex of $$N_G[u]$$ N G [ u ] . A set $$S \subseteq V$$ S ⊆ V is a ve-dominating set if for all edges $$e\in E$$ e ∈ E , there exists a vertex $$u \in S$$ u ∈ S such that u ve-dominates e. Lewis (Vertex-edge and edge-vertex parameters in graphs. Ph.D. thesis, Clemson, SC, USA, 2007) proposed a linear time algorithm for ve-domination problem for trees. In this paper, we have constructed an example where the algorithm proposed by Lewis, fails. We have proposed linear time algorithms for ve-domination and independent ve-domination problem in block graphs, which is a superclass of trees. We have also proposed a linear time algorithm for weighted ve-domination problem in trees. We have also proved that finding minimum ve-dominating set is NP-complete for undirected path graphs. Finally, we have characterized the trees with equal ve-domination and independent ve-domination number.

Keywords: Vertex-edge domination; Independent vertex-edge domination; NP-completeness (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-021-00832-z

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