65 research outputs found
Structure and algorithms for (cap, even hole)-free graphs
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48
On the structure of Gröbner bases for graph coloring ideals
In this thesis, we look at a well-known connection between the graph coloring problem and the solvability of certain systems of polynomial equations. In particular, we examine the connection between the structure of a graph and the structure of the Gröbner bases of the graph’s coloring ideal.
From a theoretical viewpoint, we show some properties of such Gröbner bases, and we develop a polynomial-time algorithm to compute a Gröbner basis for chordal graphs. From the experimental side, we state results about specific Gröbner bases and about the Gröbner fan for a variety of graph families. Moreover, some heuristics and techniques are explored that reduce the computational complexity.
The relevance of heuristic methods is justified by a section about expected intrinsic hardness of Gröbner basis computations
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
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
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