Temat: k -resolving set - Prolib Integro

Skocz do pozycji: 1.

Tytuł:: Error-Correcting Codes from k -Resolving Sets
Autorzy:: Bailey, Robert F.
Yero, Ismael G.
Tematy:: error-correcting code
k -resolving set
k -metric dimension
covering design
uncovering
grid graph; Pokaż więcej
Wydawca:: Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Powiązania:: https://bibliotekanauki.pl/articles/31343451.pdf Link otwiera się w nowym oknie
Opis:: We demonstrate a construction of error-correcting codes from graphs by means of k-resolving sets, and present a decoding algorithm which makes use of covering designs. Along the way, we determine the k-metric dimension of grid graphs (i.e., Cartesian products of paths).
Dostawca treści:: Biblioteka Nauki

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Skocz do pozycji: 2.

Tytuł:: A study of a combination of distance domination and resolvability in graphs
Autorzy:: Retnowardani, Dwi Agustin
Utoyo, Mohammad
Dafik
Susilowati, Liliek
Dliou, Kamal
Tematy:: resolving set
metric dimension
distance k-domination
distance k-resolving domination; Pokaż więcej
Wydawca:: Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Powiązania:: https://bibliotekanauki.pl/articles/59898635.pdf Link otwiera się w nowym oknie
Opis:: For $k \geq 1$, in a graph $G=(V,E)$, a set of vertices $D$ is a distance $k$-dominating set of $G$, if any vertex in $V\setminus D$ is at distance at most $k$ from some vertex in $D$. The minimum cardinality of a distance $k$-dominating set of $G$ is the distance $k$-domination number, denoted by $\gamma_k(G)$. An ordered set of vertices $W=\{w_1,w_2,\ldots,w_r\}$ is a resolving set of $G$, if for any two distinct vertices $x$ and $y$ in $V\setminus W$, there exists $1\leq i\leq r$ such that $d_G(x,w_i)\ne d_G(y,w_i)$. The minimum cardinality of a resolving set of $G$ is the metric dimension of the graph $G$, denoted by $\dim(G)$. In this paper, we introduce the distance $k$-resolving dominating set which is a subset of $V$ that is both a distance $k$-dominating set and a resolving set of $G$. The minimum cardinality of a distance $k$-resolving dominating set of $G$ is called the distance $k$-resolving domination number and is denoted by $\gamma_k^r (G)$. We give several bounds for $\gamma_k^r(G)$, some in terms of the metric dimension $\dim(G)$ and the distance $k$-domination number $\gamma_k(G)$. We determine $\gamma_k^r(G)$ when $G$ is a path or a cycle. Afterwards, we characterize the connected graphs of order $n$ having $\gamma_k^r (G)$ equal to $1$, $n-2$, and $n-1$, for $k\geq 2$. Then, we construct graphs realizing all the possible triples $(\dim(G),\gamma_k(G),\gamma_k^r (G))$, for all $k\geq 2$. Later, we determine the maximum order of a graph $G$ having distance $k$-resolving domination number $\gamma_k^r(G)=\gamma_k^r \geq 1$, we provide graphs achieving this maximum order for any positive integers $k$ and $\gamma_k^r $. Then, we establish Nordhaus-Gaddum bounds for $\gamma_k^r (G)$, for $k\geq 2$. Finally, we give relations between $\gamma_k^r(G)$ and the $k$-truncated metric dimension of graphs and give some directions for future work.
Dostawca treści:: Biblioteka Nauki

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