scholarly journals Common Independence in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1411
Author(s):  
Magda Dettlaff ◽  
Magdalena Lemańska ◽  
Jerzy Topp
Keyword(s):  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Subset ◽  
Block Graphs ◽  
Independent Domination ◽  
The Common ◽  
Independent Dominating Set

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

Independent domination number in Cayley digraphs of rectangular groups

2018 ◽  
Vol 10 (02) ◽  
pp. 1850024
Author(s):  
Nuttawoot Nupo ◽  
Sayan Panma
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Left Zero ◽  
Zero Semigroup ◽  
Left Zero Semigroup ◽  
Group A ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Rectangular Group

Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals Independent Domination Stable Trees and Unicyclic Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 820
Author(s):  
Pu Wu ◽  
Huiqin Jiang ◽  
Sakineh Nazari-Moghaddam ◽  
Seyed Mahmoud Sheikholeslami ◽  
Zehui Shao ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Basic Properties ◽  
Independent Domination ◽  
Independent Dominating Set

A set S ⊆ V ( G ) in a graph G is a dominating set if every vertex of G is either in S or adjacent to a vertex of S . A dominating set S is independent if any pair of vertices in S is not adjacent. The minimum cardinality of an independent dominating set on a graph G is called the independent domination number i ( G ) . A graph G is independent domination stable if the independent domination number of G remains unchanged under the removal of any vertex. In this paper, we study the basic properties of independent domination stable graphs, and we characterize all independent domination stable trees and unicyclic graphs. In addition, we establish bounds on the order of independent domination stable trees.

scholarly journals On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

VERTEX VULNERABILITY PARAMETER OF GEAR GRAPHS

2011 ◽  
Vol 22 (05) ◽  
pp. 1187-1195 ◽  
Author(s):  
AYSUN AYTAC ◽  
TUFAN TURACI
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Minimum Cardinality ◽  
Independent Domination

For a vertex v of a graph G = (V,E), the independent domination number (also called the lower independence number) iv(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average lower independence number of G is [Formula: see text]. In this paper, this parameter is defined and examined, also the average lower independence number of gear graphs is considered. Then, an algorithm for the average lower independence number of any graph is offered.

scholarly journals Global Alliances and Independent Domination in Some Classes of Graphs

10.37236/847 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Odile Favaron
Keyword(s):  
Bipartite Graph ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Independent Domination

A dominating set $S$ of a graph $G$ is a global (strong) defensive alliance if for every vertex $v\in S$, the number of neighbors $v$ has in $S$ plus one is at least (greater than) the number of neighbors it has in $V\setminus S$. The dominating set $S$ is a global (strong) offensive alliance if for every vertex $v\in V\setminus S$, the number of neighbors $v$ has in $S$ is at least (greater than) the number of neighbors it has in $V\setminus S$ plus one. The minimum cardinality of a global defensive (strong defensive, offensive, strong offensive) alliance is denoted by $\gamma_a(G)$ ($\gamma_{\hat a}(G)$, $\gamma_o(G)$, $\gamma_{\hat o}(G))$. We compare each of the four parameters $\gamma_a, \gamma_{\hat a}, \gamma_o, \gamma_{\hat o}$ to the independent domination number $i$. We show that $i(G)\le \gamma ^2_a(G)-\gamma_a(G)+1$ and $i(G)\le \gamma_{\hat{a}}^2(G)-2\gamma_{\hat{a}}(G)+2$ for every graph; $i(G)\le \gamma ^2_a(G)/4 +\gamma_a(G)$ and $i(G)\le \gamma_{\hat{a}}^2(G)/4 +\gamma_{\hat{a}}(G)/2$ for every bipartite graph; $i(G)\le 2\gamma_a(G)-1$ and $i(G)=3\gamma_{\hat{a}}(G)/2 -1$ for every tree and describe the extremal graphs; and that $\gamma_o(T)\le 2i(T)-1$ and $i(T)\le \gamma_{\hat o}(T)-1$ for every tree. We use a lemma stating that $\beta(T)+2i(T)\ge n+1$ in every tree $T$ of order $n$ and independence number $\beta(T)$.

Independent domination in signed graphs

2020 ◽  
pp. 2150065
Author(s):  
P. Jeyalakshmi ◽  
K. Karuppasamy ◽  
S. Arockiaraj
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Independent Domination ◽  
Independent Dominating Set

Let [Formula: see text] be a signed graph. A dominating set [Formula: see text] is said to be an independent dominating set of [Formula: see text] if [Formula: see text] is a fully negative. In this paper, we initiate a study of this parameter. We also establish the bounds and characterization on the independent domination number of a signed graph.

Strong vb-dominating and vb-independent sets of a graph

2019 ◽  
Vol 12 (01) ◽  
pp. 2050002 ◽  
Author(s):  
Sayinath Udupa ◽  
R. S. Bhat
Keyword(s):  
Lower Bounds ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Number Formula

Let [Formula: see text] be a graph. A vertex [Formula: see text] strongly (weakly) b-dominates block [Formula: see text] if [Formula: see text] ([Formula: see text]) for every vertex [Formula: see text] in the block [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vb-dominating set (SVBD-set) (WVBD-set) if every block in [Formula: see text] is strongly (weakly) b-dominated by some vertex in [Formula: see text]. The strong (weak) vb-domination number [Formula: see text] ([Formula: see text]) is the order of a minimum SVBD (WVBD) set of [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vertex block independent set (SVBI-set (WVBI-set)) if [Formula: see text] is a vertex block independent set and for every vertex [Formula: see text] and every block [Formula: see text] incident on [Formula: see text], there exists a vertex [Formula: see text] in the block [Formula: see text] such that [Formula: see text] ([Formula: see text]). The strong (weak) vb-independence number [Formula: see text] ([Formula: see text]) is the cardinality of a maximum strong (weak) vertex block independent set (SVBI-set) (WVBI-set) of [Formula: see text]. In this paper, we investigate some relationships between these four parameters. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of the bounds.

scholarly journals A lower bound on the double outer-independent domination number of a tree

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
Marcin Krzywkowski
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Domination ◽  
Independent Dominating Set

AbstractA vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph

scholarly journals Independent Domination Number in Adaptive Mesh Refinement (AMR)-WENO Scheme Networks

2020 ◽  
Vol 8 (4S5) ◽  
pp. 17-19
Keyword(s):  
Adaptive Mesh Refinement ◽  
Dominating Set ◽  
Domination Number ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Weno Scheme ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Independent Domination ◽  
Independent Dominating Set

Let G be the graph, consider the vertex set as V and edge set as E. If S is the subset of the vertex set V such that S contains vertices which has atleast one neighbor in V that is not in S, then S is said to be dominating set of G. If the vertex in S is not adjacent to one another, then S is called as the independent dominating set of G and so i(G) represents the independent domination number, the minimum cardinality of an independent dominating set in G. In this paper, we obtain independent domination number for triangular, quadrilateral, pentagonal, hexagonal, heptagonal and octagonal networks by Adaptive Mesh Refinement (AMR)-WENO Scheme.

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