36 research outputs found
Determining possible sets of leaves for spanning trees of dually chordal graphs
It will be proved that the problem of determining whether a set of vertices of a dually chordal graphs is the set of leaves of a tree compatible with it can be solved in polynomial time by establishing a connection with finding clique trees of chordal graphs with minimum number of leaves.Facultad de Ciencias Exacta
Determining possible sets of leaves for spanning trees of dually chordal graphs
It will be proved that the problem of determining whether a set of vertices of a dually chordal graphs is the set of leaves of a tree compatible with it can be solved in polynomial time by establishing a connection with finding clique trees of chordal graphs with minimum number of leaves.Facultad de Ciencias Exacta
A New Optimality Measure for Distance Dominating Sets
We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset.
The problem is a different view of the d-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the dth power of the input graph.
This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are k types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance d".
In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the d-domatic number problem. We show that the problem is NP-complete for any fixed k greater than two even when the input graph is restricted to split graphs, 2-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering k as a constant, and for strongly chordal graphs, for any k. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time O(2. 73n). Moreover, we prove that the problem cannot be approximated within ratio smaller than 2 even for split graphs, 2-connected graphs, and planar bipartite graphs of degree four. We propose a greedy 3-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work
Graph Algorithms and Applications
The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity
Upper clique transversals in graphs
A clique transversal in a graph is a set of vertices intersecting all maximal
cliques. The problem of determining the minimum size of a clique transversal
has received considerable attention in the literature. In this paper, we
initiate the study of the "upper" variant of this parameter, the upper clique
transversal number, defined as the maximum size of a minimal clique
transversal. We investigate this parameter from the algorithmic and complexity
points of view, with a focus on various graph classes. We show that the
corresponding decision problem is NP-complete in the classes of chordal graphs,
chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in
linear time in the classes of split graphs and proper interval graphs.Comment: Full version of a WG 2023 pape
New Results on Edge-coloring and Total-coloring of Split Graphs
A split graph is a graph whose vertex set can be partitioned into a clique
and an independent set. A connected graph is said to be -admissible if
admits a special spanning tree in which the distance between any two adjacent
vertices is at most . Given a graph , determining the smallest for
which is -admissible, i.e. the stretch index of denoted by
, is the goal of the -admissibility problem. Split graphs are
-admissible and can be partitioned into three subclasses: split graphs with
or . In this work we consider such a partition while dealing
with the problem of coloring a split graph. Vizing proved that any graph can
have its edges colored with or colors, and thus can be
classified as Class 1 or Class 2, respectively. When both, edges and vertices,
are simultaneously colored, i.e., a total coloring of , it is conjectured
that any graph can be total colored with or colors, and
thus can be classified as Type 1 or Type 2. These both variants are still open
for split graphs. In this paper, using the partition of split graphs presented
above, we consider the edge coloring problem and the total coloring problem for
split graphs with . For this class, we characterize Class 2 and Type
2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type
1 graph.Comment: 20 pages, 5 figure