Bounds on the vertex–edge domination number of a tree

The vertex-edge domination number of a graph, γ ve (G), is defined to be the cardinality of a smallest set D such that there exists a vertex cover C of G such that each vertex in C is dominated by a vertex in D. This is motivated by the problem of determining how many guards are needed in a graph so that a searchlight can be shone down each edge by a guard either incident to that edge or at most distance one from a vertex incident to the edge. Our main result is that for any cubic graph G with n vertices, γ ve (G) ≤ 9n/26. We also show that it is N P-hard to decide if γ ve (G) = γ(G) for bipartite graph G.

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“…The problem is formally defined below and was initially defined by Peters in [10]. The problem was also studied in [1,[7][8][9]11].…”

Section: Introductionmentioning

confidence: 99%

Dominating vertex covers: the vertex-edge domination problem

Klostermeyer

Messinger

Yeo

2021

Discuss. Math. Graph Theory

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“…A set S ⊆ V is a vertex-edge dominating set (or simply, a ve-dominating set) if for every edge e ∈ E, there exists a vertex v ∈ S such that v ve-dominates e. The minimum cardinality of a ve-dominating set of G is called the ve-domination number γ ve (G). The concept of vertex-edge domination is studied further in [2,5,6].…”

Section: Introductionmentioning

confidence: 99%

Trees with vertex-edge roman domination number twice the domination number minus one

Kumar

Venkatakrishnan

2020

Proyecciones (Antofagasta)

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A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V, E) is a function f : V (G) → {0, 1, 2} such that for each edge e = uv either max{f (u), f (v)} ≠ 0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f (w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. We characterize trees with vertexedge roman domination number equal to twice domination number minus one.

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“…In [LHHF10], the authors have characterized the trees with equal domination and vertex-edge domination number. In [KVK14], both upper and lower bounds on ve-domination number of a tree have been proved. Some upper bounds on γ ve (G) and i ve (G) and some relationship between ve-domination number and other domination parameters have been proved in [BCHH16].…”

Section: Introductionmentioning

confidence: 99%

On Vertex-Edge and Independent Vertex-Edge Domination

Shrivastava

Ranjan

2019

Combinatorial Optimization and Applications

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Given a graph G = (V, E), a vertex u ∈ V ve-dominates all edges incident to any vertex of NG [u]. A set S ⊆ V is a ve-dominating set if for all edges e ∈ E, there exists a vertex u ∈ S such that u vedominates e. Lewis [Ph.D. thesis, 2007] proposed a linear time algorithm for ve-domination problem for trees. In this paper, first we have constructed an example where the proposed algorithm fails. Then we have proposed a linear time algorithm for ve-domination problem in block graphs, which is a superclass of 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.

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Bounds on the vertex–edge domination number of a tree

Cited by 34 publications

References 3 publications

Dominating vertex covers: the vertex-edge domination problem

Dominating vertex covers: the vertex-edge domination problem

Trees with vertex-edge roman domination number twice the domination number minus one

On Vertex-Edge and Independent Vertex-Edge Domination

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