31,257 research outputs found
A characterization of trees with equal 2-domination and 2-independence numbers
A set of vertices in a graph is a -dominating set if every vertex
of not in is adjacent to at least two vertices in , and is a
-independent set if every vertex in is adjacent to at most one vertex of
. The -domination number is the minimum cardinality of a
-dominating set in , and the -independence number is the
maximum cardinality of a -independent set in . Chellali and Meddah [{\it
Trees with equal -domination and -independence numbers,} Discussiones
Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive
characterization of trees with equal -domination and -independence
numbers. Their characterization is in terms of global properties of a tree, and
involves properties of minimum -dominating and maximum -independent sets
in the tree at each stage of the construction. We provide a constructive
characterization that relies only on local properties of the tree at each stage
of the construction.Comment: 17 pages, 4 figure
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Efficient algorithms for tuple domination on co-biconvex graphs and web graphs
A vertex in a graph dominates itself and each of its adjacent vertices. The
-tuple domination problem, for a fixed positive integer , is to find a
minimum sized vertex subset in a given graph such that every vertex is
dominated by at least k vertices of this set. From the computational point of
view, this problem is NP-hard. For a general circular-arc graph and ,
efficient algorithms are known to solve it (Hsu et al., 1991 & Chang, 1998) but
its complexity remains open for . A -matrix has the consecutive
0's (circular 1's) property for columns if there is a permutation of its rows
that places the 0's (1's) consecutively (circularly) in every column.
Co-biconvex (concave-round) graphs are exactly those graphs whose augmented
adjacency matrix has the consecutive 0's (circular 1's) property for columns.
Due to A. Tucker (1971), concave-round graphs are circular-arc. In this work,
we develop a study of the -tuple domination problem on co-biconvex graphs
and on web graphs which are not comparable and, in particular, all of them
concave-round graphs. On the one side, we present an -time algorithm
for solving it for each , where is the set of universal
vertices and the total number of vertices of the input co-biconvex graph.
On the other side, the study of this problem on web graphs was already started
by Argiroffo et al. (2010) and solved from a polyhedral point of view only for
the cases and , where equals the degree of each vertex of
the input web graph . We complete this study for web graphs from an
algorithmic point of view, by designing a linear time algorithm based on the
modular arithmetic for integer numbers. The algorithms presented in this work
are independent but both exploit the circular properties of the augmented
adjacency matrices of each studied graph class.Comment: 21 pages, 7 figures. Keywords: -tuple dominating sets, augmented
adjacency matrices, stable sets, modular arithmeti
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