8 research outputs found
Measuring the vulnerability for classes of intersection graphs
AbstractA general method for the computation of various parameters measuring the vulnerability of a graph is introduced. Four measures of vulnerability are considered, i.e., the toughness, scattering number, vertex integrity and the size of a minimum balanced separator. We show how to compute these parameters by polynomial-time algorithms for various classes of intersection graphs like permutation graphs, bounded dimensional cocomparability graphs, interval graphs, trapezoid graphs and circular versions of these graph classes
Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.
We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time
A measure of graph vulnerability: scattering number
The scattering number of a graph G, denoted sc(G), is defined by sc(G)=max{c(G−S)−|S|:S⫅V(G) and c(G−S)≠1} where c(G−S) denotes the
number of components in G−S. It is one measure of graph
vulnerability. In this paper, general results on the
scattering number of a graph are considered. Firstly,
some bounds on the scattering number are given. Further,
scattering number of a binomial tree is calculated. Also
several results are given about binomial trees and
graph operations
On the Computational Complexity of Vertex Integrity and Component Order Connectivity
The Weighted Vertex Integrity (wVI) problem takes as input an -vertex
graph , a weight function , and an integer . The
task is to decide if there exists a set such that the weight
of plus the weight of a heaviest component of is at most . Among
other results, we prove that:
(1) wVI is NP-complete on co-comparability graphs, even if each vertex has
weight ;
(2) wVI can be solved in time;
(3) wVI admits a kernel with at most vertices.
Result (1) refutes a conjecture by Ray and Deogun and answers an open
question by Ray et al. It also complements a result by Kratsch et al., stating
that the unweighted version of the problem can be solved in polynomial time on
co-comparability graphs of bounded dimension, provided that an intersection
model of the input graph is given as part of the input.
An instance of the Weighted Component Order Connectivity (wCOC) problem
consists of an -vertex graph , a weight function ,
and two integers and , and the task is to decide if there exists a set
such that the weight of is at most and the weight of
a heaviest component of is at most . In some sense, the wCOC problem
can be seen as a refined version of the wVI problem. We prove, among other
results, that:
(4) wCOC can be solved in time on interval graphs,
while the unweighted version can be solved in time on this graph
class;
(5) wCOC is W[1]-hard on split graphs when parameterized by or by ;
(6) wCOC can be solved in time;
(7) wCOC admits a kernel with at most vertices.
We also show that result (6) is essentially tight by proving that wCOC cannot
be solved in time, unless the ETH fails.Comment: A preliminary version of this paper already appeared in the
conference proceedings of ISAAC 201
Structural Parameterizations of Vertex Integrity
The graph parameter vertex integrity measures how vulnerable a graph is to a
removal of a small number of vertices. More precisely, a graph with small
vertex integrity admits a small number of vertex removals to make the remaining
connected components small. In this paper, we initiate a systematic study of
structural parameterizations of the problem of computing the
unweighted/weighted vertex integrity. As structural graph parameters, we
consider well-known parameters such as clique-width, treewidth, pathwidth,
treedepth, modular-width, neighborhood diversity, twin cover number, and
cluster vertex deletion number. We show several positive and negative results
and present sharp complexity contrasts.Comment: 21 pages, 6 figures, WALCOM 202
Measuring the Vulnerability for Classes of Intersection Graphs
A general method for the computation of various parameters measuring the vulnerability of a graph is introduced. Four measures of vulnerability are considered, i.e., the toughness, scattering number, vertex integrity and the size of a minimum balanced separator. We show how to compute these parameters by polynomial time algorithms for various classes of intersection graphs like permutation graphs, bounded dimensional cocomparability graphs, interval graphs, trapezoid graphs and circular versions of these graph classes