27 research outputs found
The VC-Dimension of Graphs with Respect to k-Connected Subgraphs
We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its -connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is -complete and that it remains
-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases and . On the positive side, we
observe it can be decided in linear time for graphs of bounded clique-width
Boundary classes for graph problems involving non-local properties
We continue the study of boundary classes for NP-hard problems and focus on seven NP-hard graph problems involving non-local properties: HAMILTONIAN CYCLE, HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE, HAMILTONIAN PATH, FEEDBACK VERTEX SET, CONNECTED VERTEX COVER, CONNECTED DOMINATING SET and GRAPH VCCON DIMENSION. Our main result is the determination of the first boundary class for FEEDBACK VERTEX SET. We also determine boundary classes for HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE and HAMILTONIAN PATH and give some insights on the structure of some boundary classes for the remaining problems
Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs
We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth
Comparing Width Parameters on Graph Classes
We study how the relationship between non-equivalent width parameters changes
once we restrict to some special graph class. As width parameters, we consider
treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence
number, whereas as graph classes we consider -subgraph-free graphs,
line graphs and their common superclass, for , of -free
graphs.
We first provide a complete comparison when restricted to
-subgraph-free graphs, showing in particular that treewidth,
clique-width, mim-width, sim-width and tree-independence number are all
equivalent. This extends a result of Gurski and Wanke (2000) stating that
treewidth and clique-width are equivalent for the class of
-subgraph-free graphs.
Next, we provide a complete comparison when restricted to line graphs,
showing in particular that, on any class of line graphs, clique-width,
mim-width, sim-width and tree-independence number are all equivalent, and
bounded if and only if the class of root graphs has bounded treewidth. This
extends a result of Gurski and Wanke (2007) stating that a class of graphs
has bounded treewidth if and only if the class of line graphs of
graphs in has bounded clique-width.
We then provide an almost-complete comparison for -free graphs,
leaving one missing case. Our main result is that -free graphs of
bounded mim-width have bounded tree-independence number. This result has
structural and algorithmic consequences. In particular, it proves a special
case of a conjecture of Dallard, Milani\v{c} and \v{S}torgel.
Finally, we consider the question of whether boundedness of a certain width
parameter is preserved under graph powers. We show that the question has a
positive answer for sim-width precisely in the case of odd powers.Comment: 31 pages, 4 figures, abstract shortened due to arXiv requirement
Sublinear Longest Path Transversals and Gallai Families
We show that connected graphs admit sublinear longest path transversals. This
improves an earlier result of Rautenbach and Sereni and is related to the
fifty-year-old question of whether connected graphs admit constant-size longest
path transversals. The same technique allows us to show that -connected
graphs admit sublinear longest cycle transversals.
We also make progress toward a characterization of the graphs such that
every connected -free graph has a longest path transversal of size . In
particular, we show that the graphs on at most vertices satisfying this
property are exactly the linear forests.
Finally, we show that if the order of a connected graph is large relative
to its connectivity and , then each
vertex of maximum degree forms a longest path transversal of size
On contact graphs of paths on a grid
In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0- CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG
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