Several Roman domination graph invariants on Kneser graphs

TL;DR

This paper investigates Roman domination invariants on Kneser graphs, providing exact values and bounds for different types of Roman domination numbers, with new results especially for the case of signed Roman domination.

Contribution

It offers exact formulas for Roman and total Roman domination numbers on Kneser graphs and establishes new bounds for signed Roman domination numbers, expanding understanding of these invariants.

Findings

Exact values for $\gamma_{R}$ and $\gamma_{tR}$ when $n \\geq k(k+1)$.

New bounds for $\gamma_{sR}$ on $K_{n,2}$ depending on $n$'s parity.

Exact values for smaller graphs using existing methods.

Abstract

This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.