• S Arumugam

      Articles written in Proceedings – Mathematical Sciences

    • On Dominator Colorings in Graphs

      S Arumugam Jay Bagga K Raja Chandrasekar

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      A dominator coloring of a graph 𝐺 is a proper coloring of 𝐺 in which every vertex dominates every vertex of at least one color class. The minimum number of colors required for a dominator coloring of 𝐺 is called the dominator chromatic number of 𝐺 and is denoted by $\chi d(G)$. In this paper we present several results on graphs with $\chi d(G)=\chi(G)$ and $\chi d(G)=\gamma(G)$ where $\chi(G)$ and $\gamma(G)$ denote respectively the chromatic number and the domination number of a graph 𝐺. We also prove that if $\mu(G)$ is the Mycielskian of 𝐺, then $\chi d(G)+1\leq\chi d(\mu(G))\leq\chi d(G)+2$.

    • Co-Roman domination in graphs

      S Arumugam Karam Ebadi Martín Manrique

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      Let $G = (V,E)$ be a graph and let $f:V\to \{0, 1, 2\}$ be a function. A vertex 𝑢 is said to be protected with respect to 𝑓 if $f(u)> 0$ or $f(u)=0$ and 𝑢 is adjacent to a vertex with positive weight. The function 𝑓 is a co-Roman dominating function (CRDF) if: (i) every vertex in 𝑉 is protected, and (ii) each $v \in V$ with $f(v) > 0$ has a neighbor $u\in V$ with $f(u)=0$ such that the function $f_{vu}: V\to \{0,1,2\}$, defined by $f_{vu}(u)=1$, $f_{vu}(v)=f(v)-1$ and $f_{vu}(x)=f(x)$ for $x\in V\backslash \{u,v\}$ has no unprotected vertex. The weight of 𝑓 is $w(f)=\Sigma_{v\in V} f(v)$. The co-Roman domination number of a graph 𝐺, denoted by $\gamma_{cr}(G)$, is the minimum weight of a co-Roman dominating function on 𝐺. In this paper we initiate a study of this parameter, present several basic results, as well as some applications and directions for further research. We also show that the decision problem for the co-Roman domination number is NP-complete, even when restricted to bipartite, chordal and planar graphs.

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