731 research outputs found
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Offensive alliances in cubic graphs
An offensive alliance in a graph is a set of vertices
where for every vertex in its boundary it holds that the
majority of vertices in 's closed neighborhood are in . In the case of
strong offensive alliance, strict majority is required. An alliance is
called global if it affects every vertex in , that is, is a
dominating set of . The global offensive alliance number
(respectively, global strong offensive alliance number
) is the minimum cardinality of a global offensive
(respectively, global strong offensive) alliance in . If has
global independent offensive alliances, then the \emph{global independent
offensive alliance number} is the minimum cardinality among
all independent global offensive alliances of . In this paper we study
mathematical properties of the global (strong) alliance number of cubic graphs.
For instance, we show that for all connected cubic graph of order ,
where
denotes the line graph of . All the above bounds are tight
- …