Jasine Babu

2-Connecting outerplanar graphs without blowing up the pathwidth

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O ( p ) . This settles an open problem raised by Biedl 1, in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two-dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl 1 for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.

Fixed-Orientation Equilateral Triangle Matching of Point Sets

Given a point set P and a class \(\mathcal{C}\) of geometric objects, \(G_\mathcal{C}(P)\) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some \(C \in \mathcal{C}\) containing both p and q but no other points from P. We study G ∇ (P) graphs where ∇ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ6 graphs and TD-Delaunay graphs.

Rainbow matchings in strongly edge-colored graphs

A rainbow matching of an edge-colored graph G is a matching in which no two edges have the same color. There have been several studies regarding the maximum size of a rainbow matching in a properly edge-colored graph G in terms of its minimum degree ? ( G ) . Wang (2011) asked whether there exists a function f such that a properly edge-colored graph G with at least f ( ? ( G ) ) vertices is guaranteed to contain a rainbow matching of size ? ( G ) . This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is f ( k ) = 4 k - 4 , for k ? 4 and f ( k ) = 4 k - 3 , for k ? 3 . Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than 4 ? ( G ) - 4 vertices. Strong edge-coloring of a graph G is a restriction of proper edge-coloring where every color class is required to be an induced matching, instead of just being a matching. In this paper, we give lower bounds for the size of a maximum rainbow matching in a strongly edge-colored graph G in terms of ? ( G ) . We show that for a strongly edge-colored graph G , if | V ( G ) | ? 2 ? 3 ? ( G ) 4 ? , then G has a rainbow matching of size ? 3 ? ( G ) 4 ? , and if | V ( G ) | < 2 ? 3 ? ( G ) 4 ? , then G has a rainbow matching of size ? | V ( G ) | 2 ? . In addition, we prove that if G is a strongly edge-colored graph that is triangle-free, then it contains a rainbow matching of size at least ? ( G ) .

Polynomial time and parameterized approximation algorithms for boxicity

The boxicity (cubicity) of a graph G, denoted by box(G) (respectively cub(G)), is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (cubes) in ℝk. The problem of computing boxicity (cubicity) is known to be inapproximable in polynomial time even for graph classes like bipartite, co-bipartite and split graphs, within an O(n0.5−e) factor for any e>0, unless NP=ZPP. We prove that if a graph G on n vertices has a clique on n−k vertices, then box(G) can be computed in time

Fixed-orientation equilateral triangle matching of point sets

n^2 2^{{O(k^2 \log k)}}

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

. Using this fact, various FPT approximation algorithms for boxicity are derived. The parameter used is the vertex (or edge) edit distance of the input graph from certain graph families of bounded boxicity - like interval graphs and planar graphs. Using the same fact, we also derive an

On Induced Colourful Paths in Triangle-free Graphs

O\left(\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right)

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

factor approximation algorithm for computing boxicity, which, to our knowledge, is the first o(n) factor approximation algorithm for the problem. We also present an FPT approximation algorithm for computing the cubicity of graphs, with vertex cover number as the parameter.

Sublinear approximation algorithms for boxicity and related problems

Given a point set P and a class C of geometric objects, G C ( P ) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C ? C containing both p and q but no other points from P. We study G ? ( P ) graphs where ? is the class of downward equilateral triangles (i.e., equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half- ? 6 graphs and TD-Delaunay graphs.The main result in our paper is that for point sets P in general position, G ? ( P ) always contains a matching of size at least ? | P | - 1 3 ? and this bound is tight. We also give some structural properties of G ? ( P ) graphs, where ? is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G ? ( P ) is simply a path. Through the equivalence of G ? ( P ) graphs with ? 6 graphs, we also derive that any ? 6 graph can have at most 5 n - 11 edges, for point sets in general position.

A constant factor approximation algorithm for boxicity of circular arc graphs

Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G . This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G -the minimum number of distinct colors occurring at edges incident to any vertex of G -denoted by ? ( G ) .Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be ? 2 ? ( G ) 3 ? . Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least ? ( G ) - 1 , if 1 ? ? ( G ) ? 7 , and at least ? 3 ? ( G ) 5 ? + 1 , if ? ( G ) ? 8 . They conjectured that the tight lower bound would be ? ( G ) - 1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if ? ( G ) ? 8 , then G contains a heterochromatic path of length at least ? 2 ? ( G ) 3 ? + 1 .In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least ? ( G ) - o ( ? ( G ) ) and if the girth of G is at least 4 log 2 ( ? ( G ) ) + 2 , then it contains a heterochromatic path of length at least ? ( G ) - 2 , which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of ? 5 ? ( G ) 6 ? and for bipartite graphs we obtain a lower bound of ? 6 ? ( G ) - 3 7 ? .In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least ? 13 ? ( G ) 17 ? . This improves the previously known ? 3 ? ( G ) 4 ? bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least ? 2 ? ( G ) 3 ? .