22,601 research outputs found
Connected Domination Critical Graphs
This thesis investigates the structure of connected domination critical graphs. The characterizations developed provide an important theoretical framework for addressing a number of difficult computational problems in the areas of operations research (for example, facility locations, industrial production systems), security, communication and wireless networks, transportation and logistics networks, land surveying and computational biology. In these application areas, the problems of interest are modelled by networks and graph parameters such as domination numbers reflect the efficiency and performance of the systems
Connected majority domination vertex critical graphs
In this article, how the removal of a single vertex from a graph G can change the Connected Majority Domination number is surveyed for any graph G. A graph is Connected Domination Critical if the removal of any vertex decreases or increases its Connected Majority Domination Number. This paper gives examples and properties of CMD vertex critical graphs. There are two types namely CVR and UVR with respect to CMD sets of a graph. Also the vertex classification. V⁰CM(G), V ⁻CM(G) and V⁺CM are studied, characterisation theorems of these vertex classification are determined
A constructive characterization of total domination vertex critical graphs
AbstractLet G be a graph of order n and maximum degree Δ(G) and let γt(G) denote the minimum cardinality of a total dominating set of a graph G. A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G−v is less than the total domination number of G. We call these graphs γt-critical. For any γt-critical graph G, it can be shown that n≤Δ(G)(γt(G)−1)+1. In this paper, we prove that: Let G be a connected γt-critical graph of order n (n≥3), then n=Δ(G)(γt(G)−1)+1 if and only if G is regular and, for each v∈V(G), there is an A⊆V(G)−{v} such that N(v)∩A=0̸, the subgraph induced by A is 1-regular, and every vertex in V(G)−A−{v} has exactly one neighbor in A
3-Factor-criticality in double domination edge critical graphs
A vertex subset of a graph is a double dominating set of if
for each vertex of , where is the set of the
vertex and vertices adjacent to . The double domination number of ,
denoted by , is the cardinality of a smallest double
dominating set of . A graph is said to be double domination edge
critical if for any edge . A double domination edge critical graph with is called --critical. A graph is
-factor-critical if has a perfect matching for each set of
vertices in . In this paper we show that is 3-factor-critical if is
a 3-connected claw-free --critical graph of odd order
with minimum degree at least 4 except a family of graphs.Comment: 14 page
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