3,854 research outputs found
Graphs with at most two moplexes
A moplex is a natural graph structure that arises when lifting Dirac's
classical theorem from chordal graphs to general graphs. However, while every
non-complete graph has at least two moplexes, little is known about structural
properties of graphs with a bounded number of moplexes. The study of these
graphs is motivated by the parallel between moplexes in general graphs and
simplicial modules in chordal graphs: Unlike in the moplex setting, properties
of chordal graphs with a bounded number of simplicial modules are well
understood. For instance, chordal graphs having at most two simplicial modules
are interval. In this work we initiate an investigation of -moplex graphs,
which are defined as graphs containing at most moplexes. Of particular
interest is the smallest nontrivial case , which forms a counterpart to
the class of interval graphs. As our main structural result, we show that the
class of connected -moplex graphs is sandwiched between the classes of
proper interval graphs and cocomparability graphs; moreover, both inclusions
are tight for hereditary classes. From a complexity theoretic viewpoint, this
leads to the natural question of whether the presence of at most two moplexes
guarantees a sufficient amount of structure to efficiently solve problems that
are known to be intractable on cocomparability graphs, but not on proper
interval graphs. We develop new reductions that answer this question negatively
for two prominent problems fitting this profile, namely Graph Isomorphism and
Max-Cut. On the other hand, we prove that every connected -moplex graph
contains a Hamiltonian path, generalising the same property of connected proper
interval graphs. Furthermore, for graphs with a higher number of moplexes, we
lift the previously known result that graphs without asteroidal triples have at
most two moplexes to the more general setting of larger asteroidal sets
A polynomial algorithm for the k-cluster problem on interval graphs
This paper deals with the problem of finding, for a given graph and a given
natural number k, a subgraph of k nodes with a maximum number of edges. This
problem is known as the k-cluster problem and it is NP-hard on general graphs
as well as on chordal graphs. In this paper, it is shown that the k-cluster
problem is solvable in polynomial time on interval graphs. In particular, we
present two polynomial time algorithms for the class of proper interval graphs
and the class of general interval graphs, respectively. Both algorithms are
based on a matrix representation for interval graphs. In contrast to
representations used in most of the previous work, this matrix representation
does not make use of the maximal cliques in the investigated graph.Comment: 12 pages, 5 figure
The Neighborhood Polynomial of Chordal Graphs
The neighborhood polynomial of a graph is the generating function of
subsets of vertices in that have a common neighbor. In this paper we study
the neighborhood polynomial and the complexity of its computation for chordal
graphs. We will show that it is \NP-hard to compute the neighborhood polynomial
on general chordal graphs. Furthermore we will introduce a parameter for
chordal graphs called anchor width and an algorithm to compute the neighborhood
polynomial which runs in polynomial time if the anchor width is polynomially
bounded. Finally we will show that we can bound the anchor width for chordal
comparability graphs and chordal graphs with bounded leafage. The leafage of a
chordal graphs is the minimum number of leaves in the host tree of a subtree
representation. In particular, interval graphs have leafage at most 2. This
shows that the anchor width of interval graphs is at most quadratic
Cluster Deletion on Interval Graphs and Split Related Graphs
In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (P_5-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of the algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of P_5-free chordal graphs. Although the later result arises from the already-known reduction for P_5-free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split graphs
Parameterized Complexity of Equitable Coloring
A graph on vertices is equitably -colorable if it is -colorable and
every color is used either or times.
Such a problem appears to be considerably harder than vertex coloring, being
even for cographs and interval graphs.
In this work, we prove that it is for block
graphs and for disjoint union of split graphs when parameterized by the number
of colors; and for -free interval graphs
when parameterized by treewidth, number of colors and maximum degree,
generalizing a result by Fellows et al. (2014) through a much simpler
reduction.
Using a previous result due to Dominique de Werra (1985), we establish a
dichotomy for the complexity of equitable coloring of chordal graphs based on
the size of the largest induced star.
Finally, we show that \textsc{equitable coloring} is when
parameterized by the treewidth of the complement graph
Algorithms and Complexity for Geodetic Sets on Planar and Chordal Graphs
We study the complexity of finding the geodetic number on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that MGS remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that MGS is fixed parameter tractable for inputs of this class when parameterized by their treewidth (which equals the clique number minus one). This implies a linear-time algorithm for k-trees, for fixed k. Then, we show that MGS is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure
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
Some remarks on the geodetic number of a graph
A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs
- âŠ