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Graphs with Minimum Fractional Domatic Number [PDF]
Discrete Applied Mathematics, 2023 The domatic number of a graph is the maximum number of vertex disjoint dominating sets that partition the vertex set of the graph. In this paper we consider the fractional variant of this notion. Graphs with fractional domatic number 1 are exactly the graphs that contain an isolated vertex. Furthermore, it is known that all other graphs have fractional
Maximilien Gadouleau +3 more
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Upper domatic number of regular graphs [PDF]
Discrete Mathematics Letters, 2021 Summary: A partition \(\pi =\{V_1, V_2, \dots, V_k\}\) of the vertex set \(V(G)\) of a graph \(G= (V, E)\) is an upper domatic partition if \(V_i\) dominates \(V_j\) or \(V_j\) dominates \(V_i\) or both for all \(V_i, Vj \in \pi\). The maximum order of an upper domatic partition of \(G\) is called the upper domatic number \(D(G)\) of \(G\).
Libin Chacko Samuel, Mayamma Joseph
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The B-Domatic Number of a Graph
Discussiones Mathematicae Graph Theory, 2013 Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such ...
Favaron Odile
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The signed Roman domatic number of a digraph [PDF]
Electronic Journal of Graph Theory and Applications, 2015 Let $D$ be a finite and simple digraph with vertex set $V(D)$.A {\em signed Roman dominating function} on the digraph $D$ isa function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that$\sum_{u\in N^-[v]}f(u)\ge 1$ for every $v\in V(D)$, where $N^-[v ...
Seyed Mahmoud Sheikholeslami +1 more
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The Double Roman Domatic Number of a Digraph
Discussiones Mathematicae Graph Theory, 2020 A double Roman dominating function on a digraph D with vertex set V (D) is defined in [G. Hao, X. Chen and L. Volkmann, Double Roman domination in digraphs, Bull. Malays. Math. Sci. Soc. (2017).] as a function f : V (D) → {0, 1, 2, 3} having the property
Volkmann Lutz
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Tree domatic number in graphs [PDF]
Opuscula Mathematica, 2007 A dominating set \(S\) in a graph \(G\) is a tree dominating set of \(G\) if the subgraph induced by \(S\) is a tree. The tree domatic number of \(G\) is the maximum number of pairwise disjoint tree dominating sets in \(V(G)\).
Xue-gang Chen
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New results on upper domatic number of graphs
Communications in Combinatorics and Optimization, 2020 For a graph G = (V, E), a partition π = {V1, V2, . . . , Vk} of the vertex set V is an upper domatic partition if Vi dominates Vj or Vj dominates Vi or both for every Vi, Vj ∈ π, whenever i 6= j.
Libin Chacko Samuel, Mayamma Joseph
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Total Italian domatic number of graphs [PDF]
Computer Science Journal of Moldova, 2023 Let $G$ be a graph with vertex set $V(G)$. An \textit{Italian dominating function} (IDF) on a graph $G$ is a function $f:V(G)\longrightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to a vertex $u$ with $f(u)=2$ or to two ...
Seyed Mahmoud Sheikholeslami +1 more
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The upper domatic number of a graph [PDF]
AKCE International Journal of Graphs and Combinatorics, 2020 Let be a graph. For two disjoint sets of vertices and , set dominates set if every vertex in is adjacent to at least one vertex in . In this paper we introduce the upper domatic number , which equals the maximum order of a vertex partition such that for ...
Teresa W. Haynes +4 more
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Some new results on the b-domatic number of graphs [PDF]
Electronic Journal of Graph Theory and Applications, 2021 A domatic partition P of a graph G=(V,E) is a partition of V into classes that are pairwise disjoint dominating sets. Such a partition P is called b-maximal if no larger domatic partition P' can be obtained by gathering subsets of some classes of P to ...
Mohamed Benattalah +2 more
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