Edge Clique Cover of Claw-free Graphs

The smallest number of cliques, covering all edges of a graph $G$, is called the (edge) clique cover number of $G$ and is denoted by $cc(G)$. It is an easy observation that for every line graph $G$ with $n$ vertices, $cc(G)\leq n$. 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 $G$ is a connected claw-free graph on $n$ vertices with $\alpha(G)\geq 3$, then $cc(G)\leq n$ and equality holds if and only if $G$ is either the graph of icosahedron, or the complement of a graph on $10$ vertices called twister or the $p^{th}$ power of the cycle $C_n$, for $1\leq p \leq \lfloor (n-1)/3\rfloor$.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{$k$-Sparsest Cut} problem ($k$SC) in which, given an undirected graph $G = (V, E)$ and a parameter $k$, the objective is to partition vertex set into $k$ subsets whose maximum edge expansion is minimized. Herein, the edge expansion of a subset $S \subseteq V$ is defined as the sum of the weights of edges exiting $S$ divided by the number of vertices in $S$. Another problem that has been investigated is \textsc{$k$-Small-Set Expansion} problem ($k$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 $k$SC and $k$SSE by inspecting their parameterized complexity. On the positive side, we present two FPT algorithms for both $k$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 $k$SC problem where $k \geq 2$ 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 $k$SSE when parameterized by $k$ 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 $k,\tau\geq 3$, the problem $k$SC is NP-hard for graphs with vertex cover number at most $\tau$. We also show that $k$SC is W[1]-hard when parameterized by the treewidth of the input graph and the number~$k$ of components combined using a reduction from \textsc{Unary Bin Packing}. Furthermore, we prove that $k$SC remains NP-hard for graphs with maximum degree three and also graphs with degeneracy two. Finally, we prove that the unweighted $k$SSE is W[1]-hard for the parameter $k$

On nodal domains of finite reversible Markov processes and spectral decomposition of cycles

24 pagesInternational audienceLet $L$ be a reversible Markovian generator on a finite set $V$. Relations between the spectral decomposition of $L$ and subpartitions of the state space $V$ 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 $\mathbf{Z}_N$, 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