On a problem of E. Prisner concerning the biclique operator

Bohdan Zelinka

Displaying similar documents to “On a problem of E. Prisner concerning the biclique operator”

Signed domination and signed domatic numbers of digraphs

Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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Let D be a finite and simple digraph with the vertex set V(D), and let f:V(D) → -1,1 be a two-valued function. If $\sum_{x \in N ¯ [v]} f (x) \geq 1$ for each v ∈ V(D), where N¯[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V(D)) is called the weight w(f) of f. The minimum of weights w(f), taken over all signed dominating functions f on D, is the signed domination number $γ_{S} (D)$ of D. A set $f ₁, f ₂, . . ., f_{d}$ of signed dominating functions on D with the property that...

The classification of finite groups by using iteration digraphs

Uzma Ahmad, Muqadas Moeen (2016)

Czechoslovak Mathematical Journal

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A digraph is associated with a finite group by utilizing the power map $f : G \to G$ defined by $f (x) = x^{k}$ for all $x \in G$ , where $k$ is a fixed natural number. It is denoted by $γ_{G} (n, k)$ . In this paper, the generalized quaternion and $2$ -groups are studied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a $2$ -group are determined for a $2$ -group to be a generalized quaternion group. Further, the classification of two generated $2$ -groups as abelian...

Self-diclique circulant digraphs

Marietjie Frick, Bernardo Llano, Rita Zuazua (2015)

Mathematica Bohemica

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We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let $D = (V, A)$ be a reflexive digraph (or network). Consider $X$ and $Y$ (not necessarily disjoint) nonempty subsets of vertices (or nodes) of $D$ . A disimplex $K (X, Y)$ of $D$ is...

On the heterochromatic number of circulant digraphs

Hortensia Galeana-Sánchez, Víctor Neumann-Lara (2004)

Discussiones Mathematicae Graph Theory

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The heterochromatic number hc(D) of a digraph D, is the minimum integer k such that for every partition of V(D) into k classes, there is a cyclic triangle whose three vertices belong to different classes. For any two integers s and n with 1 ≤ s ≤ n, let $D_{n, s}$ be the oriented graph such that $V (D_{n, s})$ is the set of integers mod 2n+1 and $A (D_{n, s}) = (i, j) : j - i \in 1, 2, . . ., n ∖ s . .$ In this paper we prove that $h c (D_{n, s}) \leq 5$ for n ≥ 7. The bound is tight since equality holds when s ∈ n,[(2n+1)/3].

The total {k}-domatic number of digraphs

Seyed Mahmoud Sheikholeslami, Lutz Volkmann (2012)

Discussiones Mathematicae Graph Theory

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For a positive integer k, a total k-dominating function of a digraph D is a function f from the vertex set V(D) to the set 0,1,2, ...,k such that for any vertex v ∈ V(D), the condition $\sum_{u \in N^{-} (v)} f (u) \geq k$ is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set $f ₁, f ₂, . . ., f_{d}$ of total k-dominating functions of D with the property that $\sum_{i = 1}^{d} f_{i} (v) \leq k$ for each v ∈ V(D), is called a total k-dominating family (of functions) on D. The maximum number of functions in a total k-dominating family on D is...

On the tree structure of the power digraphs modulo $n$

Amplify Sawkmie, Madan Mohan Singh (2015)

Czechoslovak Mathematical Journal

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For any two positive integers $n$ and $k \geq 2$ , let $G (n, k)$ be a digraph whose set of vertices is ${0, 1, ..., n - 1}$ and such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^{k} \equiv b (mod n)$ . Let $n = \prod_{i = 1}^{r} p_{i}^{e_{i}}$ be the prime factorization of $n$ . Let $P$ be the set of all primes dividing $n$ and let $P_{1}, P_{2} \subseteq P$ be such that $P_{1} \cup P_{2} = P$ and $P_{1} \cap P_{2} = \emptyset$ . A fundamental constituent of $G (n, k)$ , denoted by $G_{P_{2}}^{*} (n, k)$ , is a subdigraph of $G (n, k)$ induced on the set of vertices which are multiples of $\prod_{p_{i} \in P_{2}} p_{i}$ and are relatively prime to all primes $q \in P_{1}$ . L. Somer and M. Křížek proved that the trees attached...

The Dichromatic Number of Infinite Families of Circulant Tournaments

Nahid Javier, Bernardo Llano (2017)

Discussiones Mathematicae Graph Theory

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The dichromatic number dc(D) of a digraph D is defined to be the minimum number of colors such that the vertices of D can be colored in such a way that every chromatic class induces an acyclic subdigraph in D. The cyclic circulant tournament is denoted by [...] T=C→2n+1(1,2,…,n) $T = {\vec{C}}_{2 n + 1} (1, 2, ..., n)$ , where V (T) = ℤ2n+1 and for every jump j ∈ 1, 2, . . . , n there exist the arcs (a, a + j) for every a ∈ ℤ2n+1. Consider the circulant tournament [...] C→2n+1〈k〉 ${\vec{C}}_{2 n + 1} 〈k〉$ obtained from the cyclic tournament by reversing...

Full domination in graphs

Robert C. Brigham, Gary Chartrand, Ronald D. Dutton, Ping Zhang (2001)

Discussiones Mathematicae Graph Theory

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For each vertex v in a graph G, let there be associated a subgraph $H_{v}$ of G. The vertex v is said to dominate $H_{v}$ as well as dominate each vertex and edge of $H_{v}$ . A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number $γ_{F H} (G)$ . A full dominating set of G of cardinality $γ_{F H} (G)$ is called a $γ_{F H}$ -set of G. We study three types of full domination in...

The vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_{D} [S] = ⋃_{x, y \in S} I_{D} [x, y]$ . A set S of vertices is a detour convex set if $I_{D} [S] = S$ . The detour convex hull ${[S]}_{D}$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

A note on arc-disjoint cycles in tournaments

Jan Florek (2014)

Colloquium Mathematicae

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We prove that every vertex v of a tournament T belongs to at least $m a x m i n δ ⁺ (T), 2 δ ⁺ (T) - d ⁺_{T} (v) + 1, m i n δ ¯ (T), 2 δ ¯ (T) - d ¯_{T} (v) + 1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d ⁺_{T} (v)$ (or $d ¯_{T} (v)$ ) is the out-degree (resp. in-degree) of v.

2-factors in claw-free graphs with locally disconnected vertices

Mingqiang An, Liming Xiong, Runli Tian (2015)

Czechoslovak Mathematical Journal

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An edge of $G$ is singular if it does not lie on any triangle of $G$ ; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G [N (u)]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such...