5,602 research outputs found
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
Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs
for the class of interval graphs. We give a linear-time algorithm to find one
in any graph that is not an interval graph. Tucker characterized the minimal
forbidden submatrices of binary matrices that do not have the consecutive-ones
property. We give a linear-time algorithm to find one in any binary matrix that
does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th
International Workshop on Graph-Theoretic Concepts in Computer Scienc
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
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