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2004•Mathematica ApplicataRequires access

On Graphs with Equal 2-domination and Connected 2-domination Numbers

Hongquan Yu

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Abstract

For a graph G=(V,E),a subset SV(G) is said to be a 2-dominating set of G if for each vertex u∈V-S,there exists a vertex v∈S such that d(u,v)≤2.The minimum cardinality of 2-dominating sets of G is called the 2-domination number of G and denoted by γ 2(G).A 2-dominating set S is called a connected 2-dominating set of G,if the induced subgraph 〈S〉 is connected,the connected 2-domination number of G,denoted by γc 2(G),is the minimum cardinality of connected 2-dominating sets of G.In this paper,we characterize the class of trees and unicyclic graphs for which the 2-domination numbers are equal to the connected 2-domination numbers.

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For a graph G=(V,E),a subset SV(G) is said to be a 2-dominating set of G if for each vertex u∈V-S,there exists a vertex v∈S such that d(u,v)≤2.The minimum cardinality of 2-dominating sets of G is called the 2-domination number of G and denoted by γ 2(G).A 2-dominating set S is called a connected 2-dominating set of G,if the induced subgraph 〈S〉 is connected,the connected 2-domination number of G,denoted by γc 2(G),is the minimum cardinality of connected 2-dominating sets of G.In this paper,we characterize the class of trees and unicyclic graphs for which the 2-domination numbers are equal to the connected 2-domination numbers.

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Available abstract

For a graph G=(V,E),a subset SV(G) is said to be a 2-dominating set of G if for each vertex u∈V-S,there exists a vertex v∈S such that d(u,v)≤2.The minimum cardinality of 2-dominating sets of G is called the 2-domination number of G and denoted by γ 2(G).A 2-dominating set S is called a connected 2-dominating set of G,if the induced subgraph 〈S〉 is connected,the connected 2-domination number of G,denoted by γc 2(G),is the minimum cardinality of connected 2-dominating sets of G.In this paper,we characterize the class of trees and unicyclic graphs for which the 2-domination numbers are equal to the connected 2-domination numbers.

Key concepts: Dominating set, Combinatorics, Domination analysis, Mathematics, Vertex (graph theory), Induced subgraph, Graph, Connectivity

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