9 research outputs found
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure
We continue the study of -bounded graph classes, that
is, hereditary graph classes in which the treewidth can only be large due to
the presence of a large clique, with the goal of understanding the extent to
which this property has useful algorithmic implications for the Independent Set
and related problems. In the previous paper of the series [Dallard,
Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. II.
Tree-independence number], we introduced the tree-independence number, a
min-max graph invariant related to tree decompositions. Bounded
tree-independence number implies both -boundedness and
the existence of a polynomial-time algorithm for the Maximum Weight Independent
Set problem, provided that the input graph is given together with a tree
decomposition with bounded independence number.
In this paper, we consider six graph containment relations and for each of
them characterize the graphs for which any graph excluding with respect
to the relation admits a tree decomposition with bounded independence number.
The induced minor relation is of particular interest: we show that excluding
either a minus an edge or the -wheel implies the existence of a tree
decomposition in which every bag is a clique plus at most vertices, while
excluding a complete bipartite graph implies the existence of a tree
decomposition with independence number at most . Our constructive
proofs are obtained using a variety of tools, including -refined tree
decompositions, SPQR trees, and potential maximal cliques. They imply
polynomial-time algorithms for the Independent Set and related problems in an
infinite family of graph classes; in particular, the results apply to the class
of -perfectly orientable graphs, answering a question of Beisegel,
Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.Comment: 46 pages; abstract has been shortened due to arXiv requirements. A
previous arXiv post (arXiv:2111.04543) has been reorganized into two parts;
this is the second of the two part
Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs
In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph
, the class of -graphs, defined as the intersection graphs of connected
subgraphs of some subdivision of . Recently, quite a lot of research has
been devoted to understanding the tractability border for various computational
problems, such as recognition or isomorphism testing, in classes of -graphs
for different graphs . In this work we undertake this research topic,
focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and
Zeman showed, for every fixed tree , a polynomial-time algorithm recognizing
-graphs. Tucker showed a polynomial time algorithm recognizing -graphs
(circular-arc graphs). On the other hand, Chaplick at al. showed that
recognition of -graphs is -hard if contains two different cycles
sharing an edge.
The main two results of this work narrow the gap between the -hard and
cases of -graphs recognition. First, we show that recognition of
-graphs is -hard when contains two different cycles. On the other
hand, we show a polynomial-time algorithm recognizing -graphs, where is
a graph containing a cycle and an edge attached to it (-graphs are called
lollipop graphs). Our work leaves open the recognition problems of -graphs
for every unicyclic graph different from a cycle and a lollipop. Other
results of this work, which shed some light on the cases that remain open, are
as follows. Firstly, the recognition of -graphs, where is a fixed
unicyclic graph, admits a polynomial time algorithm if we restrict the input to
graphs containing particular holes (hence recognition of -graphs is probably
most difficult for chordal graphs). Secondly, the recognition of medusa graphs,
which are defined as the union of -graphs, where runs over all unicyclic
graphs, is -complete
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