Mathew C. Francis

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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)
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)
A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph G, denoted by χ ′ s (G), is the minimum number of colours needed in any strong edge colouring of G. A graph is said to be(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)
Branchiootorenal (BOR) syndrome is a variable, autosomal-dominant disorder of the first and second embryonic branchial arches, kidneys, and urinary tract. We describe the phenotype in 45 individuals, highlighting differences and similarities reported in other studies. Characteristic temporal bone findings include cochlear hypoplasia (4/5 of normal size with(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-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)
Image fusion is an extensively discussed topic for improving the information content of images. The main objective of image fusion algorithm is to combine information from multiple images of a scene. The result of image fusion is a new image which is more feasible for human and machine perception for further image processing operations such as segmentation,(More)