arXiv subjects

N Sadagopan

Publications and source records attributed to N Sadagopan.

3 recordsLinked to original sources

arXiv2025

Secure Domination in Bisplit graphs -- A Structural and algorithmic study

A dominating set $S$ of a graph $G(V,E)$ is called a \textit{secure dominating set} if each vertex $u \in V(G) \setminus S$ is adjacent to a vertex $v \in S$ such that $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The \textit{secure domination number} $\gamma_s(G)$ of $G$ is the minimum cardinality of a secure dominating set of $G$. The \textit{Minimum Secure Domination problem} is to find a secure dominating set of a graph $G$ of cardinality $\gamma_s(G)$. In this paper, the computational complexity of the secure domination problem on several graph classes is investigated. The decision version of secure domination problem was shown to be NP-complete on star(comb) convex split graphs and bisplit graphs. So we further focus on complexity analysis of secure domination problem under additional structural restrictions on bisplit graphs. In particular, by imposing chordality as a parameter, we analyse its impact on the computational status of the problem on bisplit graphs. We establish the P versus NP-C dichotomy status of secure domination problem under restrictions on cycle length within bisplit graphs. In addition, we establish that the problem is polynomial-time solvable in chain graphs. We also prove that the secure domination problem cannot be approximated for a bisplit graph within a factor of $(1-\epsilon)~ln~|V|$ for any $\epsilon > 0$, unless $NP \subseteq DTIME(|V|^{O(log~log~|V|)})$.

cs.DM↗

arXiv2022

On Convexity in Split graphs: Complexity of Steiner tree and Domination

Given a graph $G$ with a terminal set $R \subseteq V(G)$, the Steiner tree problem (STREE) asks for a set $S\subseteq V(G) \setminus R$ such that the graph induced on $S\cup R$ is connected. A split graph is a graph which can be partitioned into a clique and an independent set. It is known that STREE is NP-complete on split graphs \cite{white1985steiner}. To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique ($K$), whereas STREE is NP-complete on tree-convex split graphs with convexity on independent set ($I$). We further strengthen our NP-complete result by establishing a dichotomy which says that for unary-tree-convex split graphs (path-convex split graphs), STREE is polynomial-time solvable, and NP-complete for binary-tree-convex split graphs (comb-convex split graphs). We also show that STREE is polynomial-time solvable for triad-convex split graphs with convexity on $I$, and circular-convex split graphs. Further, we show that STREE can be used as a framework for the dominating set problem (DS) on split graphs, and hence the classical complexity (P vs NPC) of STREE and DS is the same for all these subclasses of split graphs. Furthermore, it is important to highlight that in \cite{CHLEBIK20081264}, it is incorrectly claimed that the problem of finding a minimum dominating set on split graphs cannot be approximated within $(1-ε)\ln |V(G)|$ in polynomial-time for any $ε>0$ unless NP $\subseteq$ DTIME $n^{O(\log \log n)}$. When the input is restricted to split graphs, we show that the minimum dominating set problem has $2-\frac{1}{|I|}$-approximation algorithm that runs in polynomial time.

cs.CC↗

arXiv2017

On $P_5$-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.

cs.DM↗