Perfect Elimination Orderings for Symmetric Matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.Comment: 16 pages, 3 figure

Independent Sets in Asteroidal Triple-Free Graphs

An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an O(n4 ) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain O(n4 ) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problems on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs

Independent sets in asteroidal triple-free graphs

An asteroidal triple is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an asteroidal triple. We show that there is an O(n 2 · (¯m+1)) time algorithm to compute the maximum cardinality of an independent set for AT-free graphs, where n is the number of vertices and ¯m is the number of non edges of the input graph. Furthermore we obtain O(n 2 · (¯m+1)) time algorithms to solve the INDEPENDENT DOMINATING SET and the INDEPENDENT PERFECT DOMINATING SET problem on AT-free graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally we observe that the problems CLIQUE and PARTITION INTO CLIQUES remain NP-complete when restricted to AT-free graphs

Line-distortion, Bandwidth and Path-length of a graph

We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show: - if a graph $G$ can be embedded into the line with distortion $k$, then $G$ admits a Robertson-Seymour's path-decomposition with bags of diameter at most $k$ in $G$; - for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem; - there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; - AT-free graphs and some intersection families of graphs have path-length at most 2; - for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem

On the Kernel and Related Problems in Interval Digraphs

Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to each vertex $u$ of $G$ such that $(u,v)\in E(G)$ if and only if $S_u\cap T_v\neq\emptyset$. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs.Comment: 26 pages, 3 figure

Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness

Domination and coloring are two classic problems in graph theory. The major focus of this paper is the CD-COLORING problem which combines the flavours of domination and colouring. Let $G$ be an undirected graph. A proper vertex coloring of $G$ is a $cd-coloring$ if each color class has a dominating vertex in $G$. The minimum integer $k$ for which there exists a $cd-coloring$ of $G$ using $k$ colors is called the cd-chromatic number, $\chi_{cd}(G)$. A set $S\subseteq V(G)$ is a total dominating set if any vertex in $G$ has a neighbor in $S$. The total domination number, $\gamma_t(G)$ of $G$ is the minimum integer $k$ such that $G$ has a total dominating set of size $k$. A set $S\subseteq V(G)$ is a $separated-cluster$ if no two vertices in $S$ lie at a distance 2 in $G$. The separated-cluster number, $\omega_s(G)$, of $G$ is the maximum integer $k$ such that $G$ has a separated-cluster of size $k$. In this paper, first we explore the connection between CD-COLORING and TOTAL DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on triangle-free $d$-regular graphs for each fixed integer $d\geq 3$. We also study the relationship between the parameters $\chi_{cd}(G)$ and $\omega_s(G)$. Analogous to the well-known notion of `perfectness', here we introduce the notion of `cd-perfectness'. We prove a sufficient condition for a graph $G$ to be cd-perfect (i.e. $\chi_{cd}(H)= \omega_s(H)$, for any induced subgraph $H$ of $G$) which is also necessary for certain graph classes (like triangle-free graphs). Here, we propose a generalized framework via which we obtain several exciting consequences in the algorithmic complexities of special graph classes. In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is polynomially solvable for interval graphs