Minus Domination Numbers of Directed Graphs

2011 ◽  
Vol 267 ◽  
pp. 334-337
Author(s):  
Wen Sheng Li ◽  
Hua Ming Xing
Keyword(s):  
Lower Bounds ◽  
Directed Graph ◽  
Maximum Degree ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Domination Number ◽  
Directed Cycle ◽  
Domination Numbers

The concept of minus domination number of an undirected graph is transferred to directed graphs. Exact values are found for the directed cycle and particular types of tournaments. Furthermore, we present some lower bounds for minus domination number in terms of the order, the maximum degree, the maximum outdegree and the minimum outdegree of a directed graph.

Twin signed k-domination numbers in directed graphs

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6367-6378
Author(s):  
Nasrin Dehgardi ◽  
Maryam Atapour ◽  
Abdollah Khodkar
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Domination Number ◽  
Signed Domination ◽  
Dominating Function ◽  
Domination Numbers

Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

scholarly journals Sparse Highly Connected Spanning Subgraphs in Dense Directed Graphs

2018 ◽  
Vol 28 (3) ◽  
pp. 423-464 ◽  
Author(s):  
DONG YEAP KANG
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Analogous Result ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Additional Term ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Polynomial Time Algorithms ◽  
Bipartite Digraph

Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,n-k. For dense strongly k-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly k-connected n-vertex digraph D contains a strongly k-connected spanning subgraph with at most kn + 800k(k + Δ(D)) edges, where Δ(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D. Here, the additional term 800k(k + Δ(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly k-connected n-vertex semicomplete digraph contains a strongly k-connected spanning subgraph with at most kn + 800k2 edges, which is essentially optimal since 800k2 cannot be reduced to the number less than k(k - 1)/2.We also prove an analogous result for strongly k-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.

scholarly journals Lower bounds on the signed domination numbers of directed graphs

2009 ◽  
Vol 309 (8) ◽  
pp. 2567-2570 ◽  
Author(s):  
H. Karami ◽  
S.M. Sheikholeslami ◽  
Abdollah Khodkar
Keyword(s):  
Lower Bounds ◽  
Directed Graphs ◽  
Signed Domination ◽  
Domination Numbers

Negative Signed Domination in Digraphs

2011 ◽  
Vol 65 ◽  
pp. 145-147
Author(s):  
Wen Sheng Li
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Domination Number ◽  
Directed Cycle ◽  
Signed Domination ◽  
Signed Domination Number

The Concept of Negative Signed Domination Number of a Directed Graph Is Introduced. Exact Values Are Found for the Directed Cycle and Particular Types of Tournaments. Furthermore, it Is Proved that the Negative Signed Domination Number May Be Arbitrarily Big for Digraphs with a Directed Hamiltonian Cycle.

scholarly journals Intrinsically knotted and 4-linked directed graphs

2018 ◽  
Vol 27 (06) ◽  
pp. 1850037 ◽  
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Undirected Graph ◽  
Directed Graphs ◽  
Edge Contraction ◽  
Special Cases ◽  
Linkless Embedding ◽  
The Relationship ◽  
Knots And Links

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph [Formula: see text] and its corresponding symmetric digraph [Formula: see text]. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is [Formula: see text] in the undirected case, but [Formula: see text] for directed graphs.

scholarly journals Directed Graphs Without Short Cycles

2009 ◽  
Vol 19 (2) ◽  
pp. 285-301 ◽  
Author(s):  
JACOB FOX ◽  
PETER KEEVASH ◽  
BENNY SUDAKOV
Keyword(s):  
Directed Graph ◽  
Absolute Constant ◽  
Simple Structure ◽  
Directed Graphs ◽  
Constant Factor ◽  
Directed Cycle ◽  
Strong Component ◽  
Feedback Arc Set ◽  
Directed Cycles

For a directed graph G without loops or parallel edges, let β(G) denote the size of the smallest feedback arc set, i.e., the smallest subset X ⊂ E(G) such that G ∖ X has no directed cycles. Let γ(G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r2, where c is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour and Sullivan.This result can also be used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn2, then for any 0 ≤ m ≤ θn − o(n) it contains a directed cycle whose length is between m and m + 6θ−1/2. Moreover, there is a constant C such that either G contains directed cycles of every length between C and θn − o(n) or it is close to a digraph G′ with a simple structure: every strong component of G′ is periodic. These results are also tight up to the constant factors.

scholarly journals Lower bounds on the Roman and independent Roman domination numbers

2016 ◽  
Vol 10 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Mustapha Chellali ◽  
Teresa Haynes ◽  
Stephen Hedetniemi
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Independent Domination ◽  
Dominating Function ◽  
Domination Numbers

A Roman dominating function (RDF) on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the sum f(V) = ?v?V f(v), and the minimum weight of a Roman dominating function f is the Roman domination number ?R(G). An RDF f is called an independent Roman dominating function (IRDF) if the set of vertices assigned positive values under f is independent. The independent Roman domination number iR(G) is the minimum weight of an IRDF on G. We show that for every nontrivial connected graph G with maximum degree ?, ?R(G)? ?+1/??(G) and iR(G) ? i(G) + ?(G)/?, where ?(G) and i(G) are, respectively, the domination and independent domination numbers of G. Moreover, we characterize the connected graphs attaining each lower bound. We give an additional lower bound for ?R(G) and compare our two new bounds on ?R(G) with some known lower bounds.

scholarly journals Twin signed Roman domination numbers in directed graphs

2016 ◽  
Vol 47 (3) ◽  
pp. 357-371 ◽  
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Asghar Bodaghli ◽  
Lutz Volkmann
Keyword(s):  
Directed Graphs ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Sharp Bounds ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Dominating Function ◽  
Domination Numbers

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin signed Roman domination number $\gamma_{sR}^*(D)$ of $D$ is the minimum weight of an TSRDF on $D$. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on $\gamma_{sR}^*(D)$. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

Total pitchfork domination and its inverse in graphs

2020 ◽  
pp. 2150038
Author(s):  
Mohammed A. Abdlhusein ◽  
Manal N. Al-Harere
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Isolated Vertex ◽  
Number Formula

New two domination types are introduced in this paper. Let [Formula: see text] be a finite, simple, and undirected graph without isolated vertex. A dominating subset [Formula: see text] is a total pitchfork dominating set if [Formula: see text] for every [Formula: see text] and [Formula: see text] has no isolated vertex. [Formula: see text] is an inverse total pitchfork dominating set if [Formula: see text] is a total pitchfork dominating set of [Formula: see text]. The cardinality of a minimum (inverse) total pitchfork dominating set is the (inverse) total pitchfork domination number ([Formula: see text]) [Formula: see text]. Some properties and bounds are studied associated with maximum degree, minimum degree, order, and size of the graph. These modified domination parameters are applied on some standard and complement graphs.

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