## Mitre C. Dourado ; Vitor S. Ponciano ; Rômulo L. O. da Silva - On the monophonic rank of a graph

dmtcs:6835 - Discrete Mathematics & Theoretical Computer Science, September 15, 2022, vol. 24, no 2 - https://doi.org/10.46298/dmtcs.6835
On the monophonic rank of a graph

Authors: Mitre C. Dourado ; Vitor S. Ponciano ; Rômulo L. O. da Silva

A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free, and line graphs in polynomial time. Furthermore, we show that this parameter can computed in polynomial time for $1$-starlike graphs, i.e., for split graphs, and that its determination is $\NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.

Volume: vol. 24, no 2
Section: Graph Theory
Published on: September 15, 2022
Accepted on: August 17, 2022
Submitted on: October 10, 2020
Keywords: Computer Science - Discrete Mathematics,Computer Science - Computational Complexity,05C85,G.2.2