319 research outputs found
A look at cycles containing specified elements of a graph
AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration
Constant tolerance intersection graphs of subtrees of a tree
AbstractA chordal graph is the intersection graph of a family of subtrees of a host tree. In this paper we generalize this. A graph G=(V,E) has an (h,s,t)-representation if there exists a host tree T of maximum degree at most h, and a family of subtrees {Sv}v∈V of T, all of maximum degree at most s, such that uv∈E if and only if |Su∩Sv|⩾t. For given h,s, and t, there exist infinitely many forbidden induced subgraphs for the class of (h,s,t)-graphs. On the other hand, for fixed h⩾s⩾3, every graph is an (h,s,t)-graph provided that we take t large enough. Under certain conditions representations of larger graphs can be obtained from those of smaller ones by amalgamation procedures. Other representability and non-representability results are presented as well
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
Planarity of Streamed Graphs
In this paper we introduce a notion of planarity for graphs that are
presented in a streaming fashion. A is a stream of
edges on a vertex set . A streamed graph is
- with respect to a positive integer window
size if there exists a sequence of planar topological drawings
of the graphs such that
the common graph is drawn the same in
and in , for . The Problem with window size asks whether a given streamed
graph is -stream planar. We also consider a generalization, where there
is an additional whose edges have to be present
during each time step. These problems are related to several well-studied
planarity problems.
We show that the Problem is NP-complete even when
the window size is a constant and that the variant with a backbone graph is
NP-complete for all . On the positive side, we provide
-time algorithms for (i) the case and (ii) all
values of provided the backbone graph consists of one -connected
component plus isolated vertices and no stream edge connects two isolated
vertices. Our results improve on the Hanani-Tutte-style -time
algorithm proposed by Schaefer [GD'14] for .Comment: 21 pages, 9 figures, extended version of "Planarity of Streamed
Graphs" (9th International Conference on Algorithms and Complexity, 2015
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
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