arXiv subjects

Renjith P

Publications and source records attributed to Renjith P.

2 recordsLinked to original sources

arXiv2026

A Fine-Grained Complexity of Co-Secure Domination for Some Subclasses of Chordal Graphs

For a connected graph $G = (V, E)$, a set $D \subseteq V$ is a co-secure dominating set if $D$ is a dominating set of $G$ and for each vertex $u \in D$ there exists a vertex $v \in V \setminus D$ such that $uv \in E$ and $(D \setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. In this article, we present bounds on the co-secure domination number (size of minimum co-secure dominating set) in some restricted 2-trees, a popular subclass of chordal graphs. We also show an interesting dichotomy that the co-secure dominating set problem is NP-complete on $K_{1,r}$-free split graphs, $r\ge 4$, whereas it is linear-time solvable on $K_{1,3}$-free split graphs.

cs.DM↗

arXiv2026

Short Cycles Decide P-versus-NPC Status ofHamiltonicity on Bisplit Graphs

A connected graph G is said to be a bisplit graph if the vertex set of G can be partitioned into a stable set and a complete bipartite graph. We establish the following dichotomy with chordality being the parameter; for chordal bisplit graphs, Hamiltonian cycle (HCYCLE) and Hamiltonian path (HPATH) problems are polynomial-time solvable, and for chordal bipartite bisplit graphs, HCYCLE (HPATH) is NP-complete. We further strengthen the result of [1] and show that HCYCLE (HPATH) is polynomial-time solvable on P5-free chordal bipartite graphs (bipartite chain graphs) and NP-complete on P10-free chordal bipartite graphs. By using our polynomial results on HCYCLE (HPATH) as a framework, we solve many variants and generalizations of HCYCLE (HPATH), which are also reported in this paper.

cs.DM↗