arXiv subjects

T. Karthick

Publications and source records attributed to T. Karthick.

21 records · Page 2Linked to original sources

arXiv2015

Weighted Independent Sets in a Subclass of $P_6$-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 $P_6$-free graphs is unknown. In this note, we show that the MWIS problem can be solved in time $O(n^3m)$ for ($P_6$, banner)-free graphs by analyzing the structure of subclasses of these class of graphs. This extends the existing results for ($P_5$, banner)-free graphs, and ($P_6$, $C_4$)-free graphs. Here, $P_t$ denotes the chordless path on $t$ vertices, and a banner is the graph obtained from a chordless cycle on four vertices by adding a vertex that has exactly one neighbor on the cycle.

cs.DM↗

arXiv2015

Weighted Efficient Domination in Classes of $P_6$-free Graphs

In a graph $G$, an efficient dominating set is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarly. The Min-WED/Max-WED is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the Min-WED/Max-WED problem is unknown for $P_6$-free graphs. In this paper, we show that the Min-WED/Max-WED problem can be solved in polynomial time for two subclasses of $P_6$-free graphs, namely for ($P_6,S_{1,1,3}$)-free graphs, and for ($P_6$, bull)-free graphs.

cs.DM↗

arXiv2014

New polynomial case for efficient domination in $P_6$-free graphs

In a graph $G$, an {\it efficient dominating set} is a subset $D$ of vertices such that $D$ is an independent set and each vertex outside $D$ has exactly one neighbor in $D$. The {\textsc{Efficient Dominating Set}} problem (EDS) asks for the existence of an efficient dominating set in a given graph $G$. The EDS is known to be $NP$-complete for $P_7$-free graphs, and is known to be polynomial time solvable for $P_5$-free graphs. However, the computational complexity of the EDS problem is unknown for $P_6$-free graphs. In this paper, we show that the EDS problem can be solved in polynomial time for a subclass of $P_6$-free graphs, namely ($P_6$, banner)-free graphs.

cs.DM↗