Manu Basavaraju

Acyclic Edge Coloring of Triangle-Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ⩽ Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ⩽ 2|V(H)|−1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ⩽ Δ + 3. Triangle-free planar graphs satisfy Property A. We infer that a′(G) ⩽ Δ + 3, if G is a triangle-free planar graph. Another class of graph which satisfies Property A is 2-fold graphs (union of two forests).

Maximum weight independent sets in hole- and dart-free graphs

The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for hole-free graphs is unknown. In this paper, we first prove that the MWIS problem for (hole, dart, gem)-free graphs can be solved in O(n^3)-time. By using this result, we prove that the MWIS problem for (hole, dart)-free graphs can be solved in O(n^4)-time. Though the MWIS problem for (hole, dart, gem)-free graphs is used as a subroutine, we also give the best known time bound for the solvability of the MWIS problem in (hole, dart, gem)-free graphs.

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.

SEPARATION DIMENSION OF BOUNDED DEGREE GRAPHS

The separation dimension of a graph

Separation Dimension of Graphs and Hypergraphs

G

Maximal Induced Matchings in Triangle‐Free Graphs

is the smallest natural number

Parameterized Algorithms to Preserve Connectivity

k

On Induced Colourful Paths in Triangle-free Graphs

for which the vertices of

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

G

Parameterized algorithms for survivable network design with uniform demands

can be embedded in