Mathew C. Francis

Boxicity and maximum degree

A d-dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d-dimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity at most 2D^2. We also conjecture that the best upper bound is linear in D.

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

AbstractAn axis-parallel k-dimensional box is a Cartesian product R1×R2×⋅⋅⋅×Rk where Ri (for 1≤i≤k) is a closed interval of the form [ai,bi] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc.A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a

Chordal Bipartite Graphs with High Boxicity

\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}

The Maximum Clique Problem in Multiple Interval Graphs

approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(Δ+2)ln n⌉ dimensions, where Δ is the maximum degree of G. This algorithm implies that box (G)≤⌈(Δ+2)ln n⌉ for any graph G. Our bound is tight up to a factor of ln n.We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm.Though our general upper bound is in terms of maximum degree Δ, we show that for almost all graphs on n vertices, their boxicity is O(davln n) where dav is the average degree.

VPG and EPG bend-numbers of Halin graphs

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.

Uniquely Restricted Matchings in Interval Graphs

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t≥3 and polynomial-time solvable when t=1. The problem is also known to be NP-complete in t-track graphs when t≥4 and polynomial-time solvable when t≤2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for MAXIMUM WEIGHTED CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.

On Induced Colourful Paths in Triangle-free Graphs

A piecewise linear simple curve in the plane made up of k + 1 line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a k -bend path. Given a graph G , a collection of k -bend paths in which each path corresponds to a vertex in G and two paths have a common point if and only if the vertices corresponding to them are adjacent in G is called a B k -VPG representation of G . Similarly, a collection of k -bend paths each of which corresponds to a vertex in G is called an B k -EPG representation of G if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in G . The VPG bend-number b v ( G ) of a graph G is the minimum k such that G has a B k -VPG representation. Similarly, the EPG bend-number b e ( G ) of a graph G is the minimum k such that G has a B k -EPG representation. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph then b v ( G ) ź 1 and b e ( G ) ź 2 . These bounds are tight. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths.

Blocking Quadruple: A New Obstruction to Circular-Arc Graphs

A matching

Obstructions to chordal circular-arc graphs of small independence number

M

Dushnik-Miller dimension of contact systems of d-dimensional boxes

in a graph