Jasine Babu

Bio: Jasine Babu is an academic researcher from Indian Institutes of Technology. The author has contributed to research in topics: Vertex cover & Chordal graph. The author has an hindex of 6, co-authored 38 publications receiving 109 citations. Previous affiliations of Jasine Babu include Motorola & Indian Institute of Science.

Fixed-orientation equilateral triangle matching of point sets

Abstract: 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.

Rainbow matchings in strongly edge-colored graphs

Abstract: 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

Abstract: 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 $n^2 2^{{O(k^2 \log k)}}$. 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 $O\left(\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right)$ 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.

Fixed-Orientation Equilateral Triangle Matching of Point Sets

Abstract: 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 $G_{\bigtriangledown}(P)$ graphs where $\bigtriangledown$ 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-$\Theta_6$ graphs and TD-Delaunay graphs. The main result in our paper is that for point sets $P$ in general position, $G_{\bigtriangledown}(P)$ always contains a matching of size at least $\lceil\frac{n-2}{3}\rceil$ and this bound cannot be improved above $\lceil\frac{n-1}{3}\rceil$. We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where $\davidsstar$ 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 $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of $G_{\davidsstar}(P)$ graphs with $\Theta_6$ graphs, we also derive that any $\Theta_6$ graph can have at most $5n-11$ edges, for point sets in general position.

On graphs whose eternal vertex cover number and vertex cover number coincide

Abstract: The eternal vertex cover problem is a variant of the classical vertex cover problem defined in terms of an infinite attacker–defender game played on a graph. In each round of the game, the defender reconfigures guards from one vertex cover to another in response to a move by the attacker. The minimum number of guards required in any winning strategy of the defender when this game is played on a graph G is the eternal vertex cover number of G , denoted by evc ( G ) . It is known that given a graph G and an integer k , checking whether evc ( G ) ≤ k is NP-hard. Further, it is known that for any graph G , mvc ( G ) ≤ evc ( G ) ≤ 2 mvc ( G ) , where mvc ( G ) is the vertex cover number of G . Though a characterization is known for graphs for which evc ( G ) = 2 mvc ( G ) , a characterization of graphs for which evc ( G ) = mvc ( G ) remained as an open problem, since 2009. We achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For biconnected chordal graphs, our characterization leads to a polynomial time algorithm for precisely determining evc ( G ) and an algorithm for determining a safe strategy for guard movement in each round of the game using only evc ( G ) guards. Though the eternal vertex cover problem is only known to be in PSPACE in general, it follows from our new characterization that the problem is in NP for locally connected graphs, a graph class which includes all biconnected internally triangulated planar graphs. We also provide reductions establishing NP-completeness of the problem for biconnected internally triangulated planar graphs. As far as we know, this is the first NP-completeness result known for the problem for any graph class.

A survey of χ‐boundedness

Abstract: If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980’s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.

A survey of $\chi$-boundedness

Abstract: If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.

Rainbow Turán Problems for Paths and Forests of Stars

Abstract: For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Tur a n number of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstra e te [ Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.

Lower bounds for boxicity

Abstract: An axis-parallel b-dimensional box is a Cartesian product R1×–2×...×Rb where Ri 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 b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree δ is at least n/2(n−δ−1). 2. Consider the G(n;p) model of random graphs. Let p ≤ 1 − 40logn/n2. Then with high probability, box(G) = Ω(np(1 − p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Ω(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and \(m \leqslant c\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) edges have boxicity Ω(m/n). 3. Let G be a connected k-regular graph on n vertices. Let λ be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is at least \(\left( {\frac{{k^2 /\lambda ^2 }} {{\log \left( {1 + k^2 /\lambda ^2 } \right)}}} \right)\left( {\frac{{n - k - 1}} {{2n}}} \right)\). 4. For any positive constant c < 1, almost all balanced bipartite graphs on 2n vertices and m≤cn2 edges have boxicity Ω(m/n).

Height-Preserving Transformations of Planar Graph Drawings

Abstract: There are numerous styles of planar graph drawings, such as straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. Given a planar drawing in one of these styles, can it be converted it to another style while keeping the height unchanged? This paper answers this question for nearly all pairs of these styles, as well as for related styles that additionally restrict edges to be y-monotone and/or vertices to be horizontal line segments. These transformations can be used to develop new graph drawing results, especially for height-optimal drawings.