2,769 research outputs found
Alliance free sets in Cartesian product graphs
Let be a graph. For a non-empty subset of vertices ,
and vertex , let denote the
cardinality of the set of neighbors of in , and let .
Consider the following condition: {equation}\label{alliancecondition}
\delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex
has at least more neighbors in than it has in . A set
that satisfies Condition (\ref{alliancecondition}) for every
vertex is called a \emph{defensive} -\emph{alliance}; for every
vertex in the neighborhood of is called an \emph{offensive}
-\emph{alliance}. A subset of vertices , is a \emph{powerful}
-\emph{alliance} if it is both a defensive -alliance and an offensive -alliance. Moreover, a subset is a defensive (an offensive or
a powerful) -alliance free set if does not contain any defensive
(offensive or powerful, respectively) -alliance. In this article we study
the relationships between defensive (offensive, powerful) -alliance free
sets in Cartesian product graphs and defensive (offensive, powerful)
-alliance free sets in the factor graphs
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
Very Cost Effective Partitions in Graphs
For a graph G=(V,E) and a set of vertices S, a vertex v in S is said to be very cost effective if it is adjacent to more vertices in V -S than in S.
A bipartition pi={S, V- S} is called very cost effective if both S and V- S are very cost effective sets. Not all graphs have a very cost effective bipartition, for example, the complete graphs of odd order do not. We consider several families of graphs G, including Cartesian products and cacti graphs, to determine whether G has a very cost effective bipartition
Client–server and cost effective sets in graphs
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Client–Server and Cost Effective Sets in Graphs
For any integer k≥0, a set of vertices S of a graph G=(V,E) is k-cost-effective if for every v∈S,|N(v)∩(V∖S)|≥|N(v)∩S|+k. In this paper we study the minimum cardinality of a maximal k-cost-effective set and the maximum cardinality of a k-cost-effective set. We obtain Gallai-type results involving the k-cost-effective and global k-offensive alliance parameters, and we provide bounds on the maximum k-cost-effective number. Finally, we consider k-cost-effective sets that are also dominating. We show that computing the k-cost-effective domination number is NP-complete for bipartite graphs. Moreover, we note that not all trees have a k-cost-effective dominating set and give a constructive characterization of those that do
Alliance Partitions in Graphs.
For a graph G=(V,E), a nonempty subset S contained in V is called a defensive alliance if for each v in S, there are at least as many vertices from the closed neighborhood of v in S as in V-S. If there are strictly more vertices from the closed neighborhood of v in S as in V-S, then S is a strong defensive alliance. A (strong) defensive alliance is called global if it is also a dominating set of G. The alliance partition number (respectively, strong alliance partition number) is the maximum cardinality of a partition of V into defensive alliances (respectively, strong defensive alliances). The global (strong) alliance partition number is defined similarly. For each parameter we give both general bounds and exact values. Our major results include exact values for the alliance partition number of grid graphs and for the global alliance partition number of caterpillars
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