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NP-completeness of some generalized hop and step domination parameters in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 11، دوره 10، شماره 1، خرداد 2025، صفحه 181-193 اصل مقاله (430.82 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28699.1676 | ||
| نویسندگان | ||
| Ghazale Asemian1؛ Nader Jafari Rad* 2؛ Abolfazl Tehranian1؛ Hamid Rasouli1 | ||
| 1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran | ||
| 2Department of Mathematics, Shahed University, Tehran, Iran | ||
| چکیده | ||
| Let $r\geq 2$. A subset $S$ of vertices of a graph $G$ is a $r$-hop independent dominating set if every vertex outside $S$ is at distance $r$ from a vertex of $S$, and for any pair $v, w\in S$, $d(v, w)\neq r$. A $r$-hop Roman dominating function ($r$HRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v \in V$ with $f(v) = 0$ there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$-step Roman dominating function ($r$SRDF) is a function $f$ on $V(G)$ with values $0,1$ and $2$ having the property that for every vertex $v$ with $f(v)=0$ or $2$, there is a vertex $u$ with $f(u)=2$ and $d(u,v)=r$. A $r$HRDF $f$ is a $r$-hop Roman independent dominating function if for any pair $v, w$ with non-zero labels under $f$, $d(v, w)\neq r$. We show that the decision problem associated with each of $r$-hop independent domination, $r$-hop Roman domination, $r$-hop Roman independent domination and $r$-step Roman domination is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs. | ||
| کلیدواژهها | ||
| dominating set؛ hop dominating set؛ step dominating set؛ hop independent set؛ hop Roman dominating function؛ hop Roman independent dominating function؛ complexity | ||
| مراجع | ||
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