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# Signed star k-domatic number of a graph

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Abstract:

Let \$G\$ be a simple graph without isolated vertices with vertex set \$V(G)\$ and edge set \$E(G)\$ and let \$k\$ be a positive integer. A function \$f:E(G)longrightarrow {-1, 1}\$ is said to be a signed star \$k\$-dominating function on \$G\$ if \$sum_{ein E(v)}f(e)ge k\$ for every vertex \$v\$ of \$G\$, where \$E(v)={uvin E(G)mid uin N(v)}\$. A set \${f_1,f_2,ldots,f_d}\$ of signed star \$k\$-dominating functions on \$G\$ with the property that \$sum_{i=1}^df_i(e)le 1\$ for each \$ein E(G)\$, is called a signed star \$k\$-dominating family (of functions) on \$G\$. The maximum number of functions in a signed star \$k\$-dominating family on \$G\$ is the signed star \$k\$-domatic number of \$G\$, denoted by \$d_{kSS}(G)\$.

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