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We prove that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planar graph and two cubes have a non-empty intersection if and only if their corresponding vertices are adjacent. Moreover, when two cubes have a non-empty intersection, they just touch… (More)

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… (More)

An axis-parallel d–dimensional box is a Cartesian product R 1 × R 2 × · · · × R d where R i (for 1 ≤ i ≤ d) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension d, such that G is representable as the intersection graph of (axis–parallel) boxes in d–dimensional space. The concept of… (More)

An axis-parallel k-dimensional box is a Cartesian product R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is rep-resentable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of… (More)

A k-dimensional box is the Cartesian product R1 × R2 × · · · × R k where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then… (More)

A unit cube in k dimensional space (or k-cube in short) is defined as the Cartesian product R1 × R2 × · · · × R k where Ri(for 1 ≤ i ≤ k) is a closed interval of the form [ai, ai + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding… (More)

A k-cube (or " a unit cube in k dimensions ") is defined as the Cartesian product R1 ×. .. × R k where Ri(for 1 ≤ i ≤ k) is an interval of the form [ai, ai + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes mapped to two vertices in G have a non-empty intersection if and only if… (More)

A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochvíl and Kuběna [2] asked the question of whether the complements of planar graphs are segment graphs. We show here that the complements of all… (More)

In a typical covering problem we are given a universe U of size n, a family S (S could be given implicitly) of size m and an integer k and the objective is to check whether there exists a subfamily S ⊆ S of size at most k satisfying some desired properties. If S is required to contain all the elements of U then it corresponds to the classical Set Cover… (More)

- Mathew C. Francis, Daniel Gonçalves, Pascal Ochem
- Algorithmica
- 2013

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… (More)