25 research outputs found
Graphs without a partition into two proportionally dense subgraphs
A proportionally dense subgraph (PDS) is an induced subgraph of a graph such that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study a partition of a graph into two proportionally dense subgraphs, namely a 2-PDS partition, with and without additional constraint of connectivity of the subgraphs. We present two infinite classes of graphs: one with graphs without a 2-PDS partition, and another with graphs that only admit a disconnected 2-PDS partition. These results answer some questions proposed by Bazgan et al. (2018)
Treewidth versus clique number. II. Tree-independence number
In 2020, we initiated a systematic study of graph classes in which the
treewidth can only be large due to the presence of a large clique, which we
call -bounded. While -bounded graph
classes are known to enjoy some good algorithmic properties related to clique
and coloring problems, it is an interesting open problem whether
-boundedness also has useful algorithmic implications for
problems related to independent sets.
We provide a partial answer to this question by means of a new min-max graph
invariant related to tree decompositions. We define the independence number of
a tree decomposition of a graph as the maximum independence
number over all subgraphs of induced by some bag of . The
tree-independence number of a graph is then defined as the minimum
independence number over all tree decompositions of . Generalizing a result
on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is
given together with a tree decomposition with bounded independence number, then
the Maximum Weight Independent Packing problem can be solved in polynomial
time.
Applications of our general algorithmic result to specific graph classes will
be given in the third paper of the series [Dallard, Milani\v{c}, and
\v{S}torgel, Treewidth versus clique number. III. Tree-independence number of
graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A
previous version of this arXiv post has been reorganized into two parts; this
is the first of the two parts (the second one is arXiv:2206.15092
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
Functionality of box intersection graphs
Functionality is a graph complexity measure that extends a variety of
parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In
the present paper, we show that functionality is bounded for box intersection
graphs in , i.e. for interval graphs, and unbounded for box
intersection graphs in . We also study a parameter known as
symmetric difference, which is intermediate between twin-width and
functionality, and show that this parameter is unbounded both for interval
graphs and for unit box intersection graphs in .Comment: 11 page
Conditions for minimally tough graphs
Katona, Solt\'esz, and Varga showed that no induced subgraph can be excluded
from the class of minimally tough graphs. In this paper, we consider the
opposite question, namely which induced subgraphs, if any, must necessarily be
present in each minimally -tough graph.
Katona and Varga showed that for any rational number , every
minimally -tough graph contains a hole. We complement this result by showing
that for any rational number , every minimally -tough graph must
contain either a hole or an induced subgraph isomorphic to the -sun for some
integer .
We also show that for any rational number , every minimally
-tough graph must contain either an induced -cycle, an induced -cycle,
or two independent edges as an induced subgraph
Conditions for minimally tough graphs
Katona, Soltész, and Varga showed that no induced subgraph can be excluded from the class of minimally tough graphs. In this paper, we consider the opposite question, namely which induced subgraphs, if any, must necessarily be present in each minimally t-tough graph.
Katona and Varga showed that for any rational number t∈(1/2,1], every minimally t-tough graph contains a hole. We complement this result by showing that for any rational number t>1, every minimally t-tough graph must contain either a hole or an induced subgraph isomorphic to the k-sun for some integer k≥3.
We also show that for any rational number t>1/2, every minimally t-tough graph must contain either an induced 4-cycle, an induced 5-cycle, or two independent edges as an induced subgraph
Colourful components in k-caterpillars and planar graphs
A connected component of a vertex-coloured graph is said to be colourful if
all its vertices have different colours. By extension, a graph is colourful if
all its connected components are colourful. Given a vertex-coloured graph
and an integer , the Colourful Components problem asks whether there exist
at most edges whose removal makes colourful and the Colourful Partition
problem asks whether there exists a partition of into at most colourful
components. In order to refine our understanding of the complexity of the
problems on trees, we study both problems on -caterpillars, which are trees
with a central path such that every vertex not in is within distance
from a vertex in . We prove that Colourful Components and Colourful
Partition are NP-complete on -caterpillars with maximum degree ,
-caterpillars with maximum degree and -caterpillars with maximum
degree . On the other hand, we show that the problems are linear-time
solvable on -caterpillars. Hence, our results imply two complexity
dichotomies on trees: Colourful Components and Colourful Partition are
linear-time solvable on trees with maximum degree if (that is,
on paths), and NP-complete otherwise; Colourful Components and Colourful
Partition are linear-time solvable on -caterpillars if , and
NP-complete otherwise. We leave three open cases which, if solved, would
provide a complexity dichotomy for both problems on -caterpillars, for every
non-negative integer , with respect to the maximum degree. We also show that
Colourful Components is NP-complete on -coloured planar graphs with maximum
degree and on -coloured planar graphs with maximum degree . Our
results answer two open questions of Bulteau et al. mentioned in [30th Annual
Symposium on Combinatorial Pattern Matching, 2019]