14 research outputs found
3-Colourability of Dually Chordal Graphs in Linear Time
A graph G is dually chordal if there is a spanning tree T of G such that any
maximal clique of G induces a subtree in T. This paper investigates the
Colourability problem on dually chordal graphs. It will show that it is
NP-complete in case of four colours and solvable in linear time with a simple
algorithm in case of three colours. In addition, it will be shown that a dually
chordal graph is 3-colourable if and only if it is perfect and has no clique of
size four
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
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Algorithmic Aspects of Secure Connected Domination in Graphs
Let be a simple, undirected and connected graph. A connected
dominating set is a secure connected dominating set of , if
for each , there exists such that
and the set is a connected dominating set
of . The minimum size of a secure connected dominating set of denoted by
, is called the secure connected domination number of .
Given a graph and a positive integer the Secure Connected Domination
(SCDM) problem is to check whether has a secure connected dominating set
of size at most In this paper, we prove that the SCDM problem is
NP-complete for doubly chordal graphs, a subclass of chordal graphs. We
investigate the complexity of this problem for some subclasses of bipartite
graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite
and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem
is to find a secure connected dominating set of minimum size in the input
graph. We propose a - approximation algorithm for MSCDS,
where is the maximum degree of the input graph and prove
that MSCDS cannot be approximated within for any unless even for
bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs
with
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author