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Conference on Algorithms and Discrete Applied Mathematics

CALDAM 2018: Algorithms and Discrete Applied Mathematics pp 1–14Cite as

Efficient Domination and Efficient Edge Domination: A Brief Survey

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10743))

Abstract

In a finite undirected graph \(G=(V,E)\), a vertex \(v \in V\) dominates itself and its neighbors in G. A vertex set \(D \subseteq V\) is an efficient dominating set (e.d.s. for short) of G if every \(v \in V\) is dominated in G by exactly one vertex of D.

The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be \(\mathbb {NP}\)-complete for bipartite graphs, for (very special) chordal graphs and for line graphs but solvable in polynomial time for many subclasses. For H-free graphs, a dichotomy of the complexity of ED has been reached.

An edge set \(M \subseteq E\) is an efficient edge dominating set (e.e.d.s. for short) of G if every \(e \in E\) is dominated in G by exactly one edge of M with respect to the line graph L(G). Thus, M is an e.e.d.s. in G if and only if M is an e.d.s. in L(G). An e.e.d.s. is called dominating induced matching in various papers.

The Efficient Edge Domination (EED) problem, which asks for the existence of an e.e.d.s. in G, is known to be \(\mathbb {NP}\)-complete even for special bipartite graphs but solvable in polynomial time for various graph classes.The problems ED and EED are based on the \(\mathbb {NP}\)-complete Exact Cover problem on hypergraphs.

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    Andreas Brandstädt

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  2. University of Calcutta, Kolkata, India

    Partha P. Goswami

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Brandstädt, A. (2018). Efficient Domination and Efficient Edge Domination: A Brief Survey. In: Panda, B., Goswami, P. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2018. Lecture Notes in Computer Science(), vol 10743. Springer, Cham. https://doi.org/10.1007/978-3-319-74180-2_1

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