On strictly Chordality-k graphs

Strictly Chordality-k graphs (SC_k graphs) are graphs which are either cycle free or every induced cycle is exactly k, for some fixed k, k \geq 3. Note that k = 3 and k = 4 are precisely the Chordal graphs and Chordal Bipartite graphs, respectively. In this paper, we initiate a structural and an algorithmic study of SCk, k \geq 5 graphs.Comment: 25 pages, 11 figures, 2 tables, 3 algorithms, In revision in Discrete Applied Mathematic

On Some Combinatorial Problems in Cographs

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs, our framework yields polynomial-time algorithms for all three problems.Comment: 21 pages, 4 figure

Hamiltonian Path in Split Graphs- a Dichotomy

In this paper, we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of $K_{1,4}$-free split graphs in the context of Hamiltonian path problem, and as a consequence, we obtain a polynomial-time algorithm to the Hamiltonian path problem in $K_{1,4}$-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in $K_{1,5}$-free split graphs by reducing from Hamiltonian cycle problem in $K_{1,5}$-free split graphs. Thus this paper establishes a "thin complexity line" separating NP-complete instances and polynomial-time solvable instances.Comment: 39 pages, 4 figures, CALDAM 201

2-Trees: Structural Insights and the study of Hamiltonian Paths

For a connected graph, a path containing all vertices is known as \emph{Hamiltonian path}. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a Hamiltonian path in general graphs is NP-Complete. We present a necessary and sufficient condition for the existence of Hamiltonian paths in 2-trees. Using our characterization, we also present a linear-time algorithm for the existence of Hamiltonian paths in 2-trees. Our characterization is based on a deep understanding of the structure of 2-trees and the combinatorics presented here may be used in other combinatorial problems restricted to 2-trees.Comment: 16 pages, 7 figure

Hamiltonicity in Convex Bipartite Graphs

For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have convexity with respect to $X$. Further, convex bipartite graphs are a subclass of chordal bipartite graphs. In this paper, we present a necessary and sufficient condition for the existence of a Hamiltonian cycle in convex bipartite graphs and further we obtain a linear-time algorithm for this graph class. We also show that Chvatal's necessary condition is sufficient for convex bipartite graphs. The closely related problem is HAMILTONIAN PATH whose complexity is open in convex bipartite graphs. We classify the class of convex bipartite graphs as {\em monotone} and {\em non-monotone} graphs. For monotone convex bipartite graphs, we present a linear-time algorithm to output a Hamiltonian path. We believe that these results can be used to obtain algorithms for Hamiltonian path problem in non-monotone convex bipartite graphs. It is important to highlight (a) in \cite{keil,esha}, it is incorrectly claimed that Hamiltonian path problem in convex bipartite graphs is polynomial-time solvable by referring to \cite{muller} which actually discusses Hamiltonian cycle (b) the algorithm appeared in \cite{esha} for the longest path problem (Hamiltonian path problem) in biconvex and convex bipartite graphs have an error and it does not compute an optimum solution always. We present an infinite set of counterexamples in support of our claim

A Characterization of all Stable Minimal Separator Graphs

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We show that such graphs are precisely those in which the induced subgraph, namely, a cycle with exactly one chord is forbidden. We also show that deciding maximum such forbidden subgraph is NP-complete by establishing a polynomial time reduction from maximum induced cycle problem [1]. This result is of independent interest and can be used in other combinatorial problems. Secondly, we prove that a graph has the following property: every minimal edge separator induces a matching (that is no two edges share a vertex in common) if and only if it is a tree

On P5P_5-free Chordal bipartite graphs

A bipartite graph is chordal bipartite if every cycle of length at least 6 has a chord in it. In this paper, we investigate the structure of $P_5$-free chordal bipartite graphs and show that these graphs have a Nested Neighborhood Ordering, a special ordering among its vertices. Further, using this ordering, we present polynomial-time algorithms for classical problems such as Hamiltonian cycle (path) and longest path. Two variants of Hamiltonian path include Steiner path and minimum leaf spanning tree, and we obtain polynomial-time algorithms for these problems as well restricted to $P_5$-free chordal bipartite graphs.Comment: Presented in ICMCE 201

Tri-connectivity Augmentation in Trees

For a connected graph, a {\em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a graph is $k$-vertex connected if its vertex connectivity is $k$, $k\geq 1$. Given a $k$-vertex connected graph $G$, the combinatorial problem {\em vertex connectivity augmentation} asks for a minimum number of edges whose augmentation to $G$ makes the resulting graph $(k+1)$-vertex connected. In this paper, we initiate the study of $r$-vertex connectivity augmentation whose objective is to find a $(k+r)$-vertex connected graph by augmenting a minimum number of edges to a $k$-vertex connected graph, $r \geq 1$. We shall investigate this question for the special case when $G$ is a tree and $r=2$. In particular, we present a polynomial-time algorithm to find a minimum set of edges whose augmentation to a tree makes it 3-vertex connected. Using lower bound arguments, we show that any tri-vertex connectivity augmentation of trees requires at least $\lceil \frac {2l_1+l_2}{2} \rceil$ edges, where $l_1$ and $l_2$ denote the number of degree one vertices and degree two vertices, respectively. Further, we establish that our algorithm indeed augments this number, thus yielding an optimum algorithm.Comment: 10 pages, 2 figures, 3 algorithms, Presented in ICGTA 201

Spanning Trees in 2-trees

A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood consists of two adjacent vertices. We use this construction order both to inductively list the spanning trees without repetition and to give bounds on the number of them. We determine the $n$-vertex $2$-trees having the most and the fewest spanning trees. The $2$-tree with the fewest is unique; it has $n-2$ vertices of degree $2$ and has $n2^{n-3}$ spanning trees. Those with the most are all those having exactly two vertices of degree $2$, and their number of spanning trees is the Fibonacci number $F_{2n-2}$.Comment: 10 Pages, 4 Figure

Listing All Spanning Trees in Halin Graphs - Sequential and Parallel view

For a connected labelled graph $G$, a {\em spanning tree} $T$ is a connected and an acyclic subgraph that spans all vertices of $G$. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of $G$. A Halin graph is a graph obtained from a tree with no degree two vertices and by joining all leaves with a cycle. We present a sequential and parallel algorithm to enumerate all spanning trees in Halin graphs. Our approach enumerates without repetitions and we make use of $O((2pd)^{p})$ processors for parallel algorithmics, where $d$ and $p$ are the depth, the number of leaves, respectively, of the Halin graph. We also prove that the number of spanning trees in Halin graphs is $O((2pd)^{p})$.Comment: 13 pages, 5 figure