132 research outputs found
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
Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five
In this paper, we address the maximum number of vertices of induced forests
in subcubic graphs with girth at least four or five. We provide a unified
approach to prove that every 2-connected subcubic graph on vertices and
edges with girth at least four or five, respectively, has an induced forest on
at least or vertices, respectively, except
for finitely many exceptional graphs. Our results improve a result of Liu and
Zhao and are tight in the sense that the bounds are attained by infinitely many
2-connected graphs. Equivalently, we prove that such graphs admit feedback
vertex sets with size at most or , respectively.
Those exceptional graphs will be explicitly constructed, and our result can be
easily modified to drop the 2-connectivity requirement
A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs
AbstractIn this paper, we consider the problem of finding a maximum weight 2-matching containing no cycle of a length of at most three in a weighted simple graph, which we call the weighted triangle-free 2-matching problem. Although the polynomial solvability of this problem is still open in general graphs, a polynomial-time algorithm is given by Hartvigsen and Li for the problem in subcubic graphs, i.e., graphs with a maximum degree of at most three. Our contribution is to provide another polynomial-time algorithm for the weighted triangle-free 2-matching problem in subcubic graphs. Our algorithm consists of two basic algorithms: a steepest ascent algorithm and a classical maximum weight2-matching algorithm, and is justified by fundamental results from the theory of discrete convex functions on jump systems
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
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