7 research outputs found
Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star
Many recent works address the question of characterizing induced obstructions
to bounded treewidth. In 2022, Lozin and Razgon completely answered this
question for graph classes defined by finitely many forbidden induced
subgraphs. Their result also implies a characterization of graph classes
defined by finitely many forbidden induced subgraphs that are
-bounded, that is, treewidth can only be large due to the presence
of a large clique. This condition is known to be satisfied for any graph class
with bounded tree-independence number, a graph parameter introduced
independently by Yolov in 2018 and by Dallard, Milani\v{c}, and \v{S}torgel in
2024. Dallard et al. conjectured that -boundedness is actually
equivalent to bounded tree-independence number. We address this conjecture in
the context of graph classes defined by finitely many forbidden induced
subgraphs and prove it for the case of graph classes excluding an induced star.
We also prove it for subclasses of the class of line graphs, determine the
exact values of the tree-independence numbers of line graphs of complete graphs
and line graphs of complete bipartite graphs, and characterize the
tree-independence number of -free graphs, which implies a linear-time
algorithm for its computation. Applying the algorithmic framework provided in a
previous paper of the series leads to polynomial-time algorithms for the
Maximum Weight Independent Set problem in an infinite family of graph classes.Comment: 26 page
Computing Subset Vertex Covers in H-Free Graphs
We consider a natural generalization of Vertex Cover: the Subset Vertex Cover problem, which is to decide for a graph G=(V,E), a subset T⊆V and integer k, if V has a subset S of size at most k, such that S contains at least one end-vertex of every edge incident to a vertex of T. A graph is H-free if it does not contain H as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP-complete on subcubic (claw,diamond)-free planar graphs and on 2-unipolar graphs, a subclass of 2P3-free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under P ≠ NP). We also prove new polynomial time results. We first give a dichotomy on graphs where G[T] is H-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs G, for which G[T] is H-free, if H=sP1+tP2 and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for (sP1+P2+P3)-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on H-free graphs
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
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
Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time
We show that the \textsc{Maximum Weight Independent Set} problem
(\textsc{MWIS}) can be solved in quasi-polynomial time on -free graphs
(graphs excluding a fixed graph as an induced subgraph) for every whose
every connected component is a path or a subdivided claw (i.e., a tree with at
most three leaves). This completes the dichotomy of the complexity of
\textsc{MWIS} in -free graphs for any finite set of
graphs into NP-hard cases and cases solvable in quasi-polynomial time, and
corroborates the conjecture that the cases not known to be NP-hard are actually
polynomial-time solvable.
The key graph-theoretic ingredient in our result is as follows. Fix an
integer . Let be the graph created from three paths on
edges by identifying one endpoint of each path into a single vertex. We
show that, given a graph , one can in polynomial time find either an induced
in , or a balanced separator consisting of \Oh(\log |V(G)|)
vertex neighborhoods in , or an extended strip decomposition of (a
decomposition almost as useful for recursion for \textsc{MWIS} as a partition
into connected components) with each particle of weight multiplicatively
smaller than the weight of . This is a strengthening of a result of Majewski
et al.\ [ICALP~2022] which provided such an extended strip decomposition only
after the deletion of \Oh(\log |V(G)|) vertex neighborhoods. To reach the
final result, we employ an involved branching strategy that relies on the
structural lemma presented above.Comment: 58 pages, 4 figure