4,635 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
Extremal problems involving forbidden subgraphs
In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs.
In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free -colorable graphs and we find the asymptotic threshold of density of -colorable graphs of large girth when .
In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of -dynamic chromatic number of graphs in terms of other parameters.
In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs.
In particular, we characterize hypergraphs with the maximum number of edges that contain no -regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure
On bounding the difference between the maximum degree and the chromatic number by a constant
We provide a finite forbidden induced subgraph characterization for the graph
class , for all , which is defined as
follows. A graph is in if for any induced subgraph, holds, where is the maximum degree and is the
chromatic number of the subgraph.
We compare these results with those given in [O. Schaudt, V. Weil, On
bounding the difference between the maximum degree and the clique number,
Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI:
10.1007/s00373-014-1468-3], where we studied the graph class , for
, whose graphs are such that for any induced subgraph,
holds, where denotes the clique number of
a graph. In particular, we give a characterization in terms of
and of those graphs where the neighborhood of every vertex is
perfect.Comment: 10 pages, 4 figure
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