Boxicity and Cubicity of Asteroidal Triple free graphs

An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as \boxi(G), is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as \cub(G), is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as \claw. Let $G$ be an AT-free graph with chromatic number $\chi(G)$ and claw number \claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2) $\leq$ \chi(G)(\ceil{\log_2 \claw} +2). If $G$ is an AT-free graph having girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq 2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of permutation graphs with a superior result. Here we consider the boxicity of AT-free graphs, which is a super class of permutation graph

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

Hadwiger number of graphs with small chordality

The Hadwiger number of a graph G is the largest integer h such that G has the complete graph K_h as a minor. We show that the problem of determining the Hadwiger number of a graph is NP-hard on co-bipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most $s$. We show that this problem can be solved in polynomial time on AT-free graphs when s>=2, but is NP-hard on chordal graphs for every fixed s>=2

Linear time algorithms for dominating pairs in asteroidal triple-free graphs

Abstract. An independent set of three vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighborhood of the third. A graph is asteroidal triplefree (AT-free) if it contains no asteroidal triple. The motivation for this investigation is provided, in part, by the fact that AT-free graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected AT-free graphs. The resulting simple algorithm, based on the well-known lexicographic breadth-first search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity O(|V | 3) for input graph G =(V,E). In addition, we indicate how our algorithm can be extended to find, in time linear in the size of the input, all dominating pairs in a connected AT-free graph with diameter greater than 3. A remarkable feature of the extended algorithm is that, even though there may be O(|V | 2) dominating pairs, the algorithm can compute and represent them in linear time