27 research outputs found
Edge Clique Cover of Claw-free Graphs
The smallest number of cliques, covering all edges of a graph , is
called the (edge) clique cover number of and is denoted by . It
is an easy observation that for every line graph with vertices,
. G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26;
MR1761707] extended this observation to all quasi-line graphs and questioned if
the same assertion holds for all claw-free graphs. In this paper, using the
celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour,
we give an affirmative answer to this question for all claw-free graphs with
independence number at least three. In particular, we prove that if is a
connected claw-free graph on vertices with , then and equality holds if and only if is either the graph of
icosahedron, or the complement of a graph on vertices called twister or
the power of the cycle , for .Comment: 74 pages, 4 figure
On the Parameterized Complexity of the Acyclic Matching Problem
A matching is a set of edges in a graph with no common endpoint. A matching M
is called acyclic if the induced subgraph on the endpoints of the edges in M is
acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for
an acyclic matching of size k in G. The problem is known to be NP-complete. In
this paper, we investigate the complexity of the problem in different aspects.
First, we prove that the problem remains NP-complete for the class of planar
bipartite graphs of maximum degree three and arbitrarily large girth. Also, the
problem remains NP-complete for the class of planar line graphs with maximum
degree four. Moreover, we study the parameterized complexity of the problem. In
particular, we prove that the problem is W[1]-hard on bipartite graphs with
respect to the parameter k. On the other hand, the problem is fixed parameter
tractable with respect to the parameters tw and (k, c4), where tw and c4 are
the treewidth and the number of cycles with length 4 of the input graph. We
also prove that the problem is fixed parameter tractable with respect to the
parameter k for the line graphs and every proper minor-closed class of graphs
(including planar graphs)
On the Parameterized Complexity of Sparsest Cut and Small-set Expansion Problems
We study the NP-hard \textsc{-Sparsest Cut} problem (SC) in which,
given an undirected graph and a parameter , the objective is to
partition vertex set into subsets whose maximum edge expansion is
minimized. Herein, the edge expansion of a subset is defined as
the sum of the weights of edges exiting divided by the number of vertices
in . Another problem that has been investigated is \textsc{-Small-Set
Expansion} problem (SSE), which aims to find a subset with minimum edge
expansion with a restriction on the size of the subset. We extend previous
studies on SC and SSE by inspecting their parameterized complexity. On
the positive side, we present two FPT algorithms for both SSE and 2SC
problems where in the first algorithm we consider the parameter treewidth of
the input graph and uses exponential space, and in the second we consider the
parameter vertex cover number of the input graph and uses polynomial space.
Moreover, we consider the unweighted version of the SC problem where is fixed and proposed two FPT algorithms with parameters treewidth and
vertex cover number of the input graph. We also propose a randomized FPT
algorithm for SSE when parameterized by and the maximum degree of the
input graph combined. Its derandomization is done efficiently.
\noindent On the negative side, first we prove that for every fixed integer
, the problem SC is NP-hard for graphs with vertex cover
number at most . We also show that SC is W[1]-hard when parameterized
by the treewidth of the input graph and the number~ of components combined
using a reduction from \textsc{Unary Bin Packing}. Furthermore, we prove that
SC remains NP-hard for graphs with maximum degree three and also graphs with
degeneracy two. Finally, we prove that the unweighted SSE is W[1]-hard for
the parameter
On nodal domains of finite reversible Markov processes and spectral decomposition of cycles
24 pagesInternational audienceLet be a reversible Markovian generator on a finite set . Relations between the spectral decomposition of and subpartitions of the state space into a given number of components which are optimal with respect to min-max or max-min Dirichlet connectivity criteria are investigated. Links are made with higher order Cheeger inequalities and with a generical characterization of subpartitions given by the nodal domains of an eigenfunction. These considerations are applied to generators whose positive rates are supported by the edges of a discrete cycle , to obtain a full description of their spectra and of the shapes of their eigenfunctions, as well as an interpretation of the spectrum through a double covering construction. Also, we prove that for these generators, higher Cheeger inequalities hold, with a universal constant factor 48