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Wyświetlanie 1-15 z 225
Tytuł:
Total domination versus paired domination
Autorzy:
Schaudt, Oliver
Powiązania:
https://bibliotekanauki.pl/articles/743224.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
upper total domination
paired domination
upper paired domination
generalized claw-free graphs
Opis:
A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γₜ. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γₜ. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by γₚ. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Γₚ.
In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound γₚ/γₜ ≤ 2 - 2/r for $K_{1,r}$-free graphs. As a consequence, we obtain the sharp bound γₚ/γₜ ≤ 2 - 2/(Δ+1), where Δ is the maximum degree. We also show for each r ≥ 2 that ${C₅,T_r}$-free graphs fulfill the sharp bound γₚ/γₜ ≤ 2 - 2/r, where $T_r$ is obtained from $K_{1,r}$ by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Γₚ/Γₜ. Further, we prove that a graph hereditarily has an induced paired dominating set if and only if γₚ ≤ Γₜ holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio γₚ/Γₜ taken over the induced subgraphs of a graph. As a consequence, we prove for each r ≥ 3 the sharp bound γₚ/Γₜ ≤ 2 - 2/r for graphs that do not contain the corona of $K_{1,r}$ as subgraph. In particular, we obtain the sharp bound γₚ/Γₜ ≤ 2 - 2/Δ.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 3; 435-447
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Secure domination and secure total domination in graphs
Autorzy:
Klostermeyer, William
Mynhardt, Christina
Powiązania:
https://bibliotekanauki.pl/articles/743322.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
secure domination
total domination
secure total domination
clique covering
Opis:
A secure (total) dominating set of a graph G = (V,E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V-X, there exists x ∈ X adjacent to u such that $(X-{x}) ∪ {u}$ is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number $γ_s(G)(γ_{st}(G))$. We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then $γ_{st}(G) ≤ γ_s(G)$. We also show that $γ_{st}(G)$ is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 267-284
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Paired-domination
Autorzy:
Fitzpatrick, S.
Hartnell, B.
Powiązania:
https://bibliotekanauki.pl/articles/744199.pdf
Data publikacji:
1998
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
paired-domination
matching
Opis:
We are interested in dominating sets (of vertices) with the additional property that the vertices in the dominating set can be paired or matched via existing edges in the graph. This could model the situation of guards or police where each has a partner or backup. This paper will focus on those graphs in which the number of matched pairs of a minimum dominating set of this type equals the size of some maximal matching in the graph. In particular, we characterize the leafless graphs of girth seven or more of this type.
Źródło:
Discussiones Mathematicae Graph Theory; 1998, 18, 1; 63-72
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Full domination in graphs
Autorzy:
Brigham, Robert
Chartrand, Gary
Dutton, Ronald
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743419.pdf
Data publikacji:
2001
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
full domination
full star domination
full closed domination
full open domination
Opis:
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 $γ_{FH}(G)$. A full dominating set of G of cardinality $γ_{FH}(G)$ is called a $γ_{FH}$-set of G. We study three types of full domination in graphs: full star domination, where $H_v$ is the maximum star centered at v, full closed domination, where $H_v$ is the subgraph induced by the closed neighborhood of v, and full open domination, where $H_v$ is the subgraph induced by the open neighborhood of v.
Źródło:
Discussiones Mathematicae Graph Theory; 2001, 21, 1; 43-62
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Domination and leaf density in graphs
Autorzy:
Pedersen, Anders
Powiązania:
https://bibliotekanauki.pl/articles/744357.pdf
Data publikacji:
2005
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
bounds
domination number
leaves
partioned domination
total domination number
Opis:
The domination number γ(G) of a graph G is the minimum cardinality of a subset D of V(G) with the property that each vertex of V(G)-D is adjacent to at least one vertex of D. For a graph G with n vertices we define ε(G) to be the number of leaves in G minus the number of stems in G, and we define the leaf density ζ(G) to equal ε(G)/n. We prove that for any graph G with no isolated vertex, γ(G) ≤ n(1- ζ(G))/2 and we characterize the extremal graphs for this bound. Similar results are obtained for the total domination number and the partition domination number.
Źródło:
Discussiones Mathematicae Graph Theory; 2005, 25, 3; 251-259
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Characterization of block graphs with equal 2-domination number and domination number plus one
Autorzy:
Hansberg, Adriana
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743677.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
2-domination
multiple domination
block graph
Opis:
Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V(G) is a p-dominating set of the graph G, if every vertex v ∈ V(G)-D is adjacent with at least p vertices of D. The p-domination number γₚ(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination number γ(G).
If G is a nontrivial connected block graph, then we show that γ₂(G) ≥ γ(G)+1, and we characterize all connected block graphs with γ₂(G) = γ(G)+1. Our results generalize those of Volkmann [12] for trees.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 1; 93-103
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Efficient (j,k)-domination
Autorzy:
Rubalcaba, Robert
Slater, Peter
Powiązania:
https://bibliotekanauki.pl/articles/743411.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
efficient domination
multiple domination
Opis:
A dominating set S of a graph G is called efficient if |N[v]∩ S| = 1 for every vertex v ∈ V(G). That is, a dominating set S is efficient if and only if every vertex is dominated exactly once. In this paper, we investigate efficient multiple domination. There are several types of multiple domination defined in the literature: k-tuple domination, {k}-domination, and k-domination. We investigate efficient versions of the first two as well as a new type of multiple domination.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 3; 409-423
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A note on domination parameters in random graphs
Autorzy:
Bonato, Anthony
Wang, Changping
Powiązania:
https://bibliotekanauki.pl/articles/743019.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
random graphs
independent domination
total domination
Opis:
Domination parameters in random graphs G(n,p), where p is a fixed real number in (0,1), are investigated. We show that with probability tending to 1 as n → ∞, the total and independent domination numbers concentrate on the domination number of G(n,p).
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 335-343
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Trees with equal restrained domination and total restrained domination numbers
Autorzy:
Raczek, Joanna
Powiązania:
https://bibliotekanauki.pl/articles/743684.pdf
Data publikacji:
2007
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total restrained domination number
restrained domination number
trees
Opis:
For a graph G = (V,E), a set D ⊆ V(G) is a total restrained dominating set if it is a dominating set and both ⟨D⟩ and ⟨V(G)-D⟩ do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V(G) is a restrained dominating set if it is a dominating set and ⟨V(G)-D⟩ does not contain an isolated vertex. The cardinality of a minimum restrained dominating set in G is the restrained domination number. We characterize all trees for which total restrained and restrained domination numbers are equal.
Źródło:
Discussiones Mathematicae Graph Theory; 2007, 27, 1; 83-91
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A remark on the (2,2)-domination number
Autorzy:
Korneffel, Torsten
Meierling, Dirk
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743033.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
distance domination number
p-domination number
Opis:
A subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter $γ_{k,p}(G)$ denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that $γ_{k,p}(G) ≤ (p/(p+k))n(G)$ for any graph G with δₖ(G) ≥ k+p-1, where the latter means that every vertex is within distance k to at least k+p-1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that $γ_{2,2}(G) ≤ (n(G)+1)/2$ for all connected graphs G and characterize all connected graphs with $γ_{2,2} = (n+1)/2$. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that δ₂ ≥ 3.
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 2; 361-366
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Some remarks on α-domination
Autorzy:
Dahme, Franz
Rautenbach, Dieter
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/744557.pdf
Data publikacji:
2004
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
α-domination
domination
Opis:
Let α ∈ (0,1) and let $G = (V_G,E_G$) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D ⊆ V_G$ is called an α-dominating set of G, if $|N_G(u) ∩ D| ≥ αd_G(u)$ for all $u ∈ V_G∖D$. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.
Źródło:
Discussiones Mathematicae Graph Theory; 2004, 24, 3; 423-430
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On stratification and domination in graphs
Autorzy:
Gera, Ralucca
Zhang, Ping
Powiązania:
https://bibliotekanauki.pl/articles/743948.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
stratified graph
F-domination
domination
Opis:
A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F-coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F-domination number $γ_F(G)$ is the minimum number of red vertices in an F-coloring of G. In this paper, we study F-domination, where F is a 2-stratified red-blue-blue path of order 3 rooted at a blue end-vertex. We present characterizations of connected graphs of order n with F-domination number n or 1 and establish several realization results on F-domination number and other domination parameters.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 2; 249-272
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total outer-connected domination in trees
Autorzy:
Cyman, Joanna
Powiązania:
https://bibliotekanauki.pl/articles/744028.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total outer-connected domination number
domination number
Opis:
Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by $γ_{tc}(G)$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then $γ_{tc}(T) ≥ ⎡2n/3⎤$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 3; 377-383
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Graphs with large double domination numbers
Autorzy:
Henning, Michael
Powiązania:
https://bibliotekanauki.pl/articles/744277.pdf
Data publikacji:
2005
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
bounds
domination
double domination
minimum degree two
Opis:
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{×2}(G)$. If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that $γ_{×2}(G) ≤ 3n/4$ and we characterize those graphs achieving equality.
Źródło:
Discussiones Mathematicae Graph Theory; 2005, 25, 1-2; 13-28
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Characterization of trees with equal 2-domination number and domination number plus two
Autorzy:
Chellali, Mustapha
Volkmann, Lutz
Powiązania:
https://bibliotekanauki.pl/articles/743587.pdf
Data publikacji:
2011
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
2-domination number
domination number
trees
Opis:
Let G = (V(G),E(G)) be a simple graph, and let k be a positive integer. A subset D of V(G) is a k-dominating set if every vertex of V(G) - D is dominated at least k times by D. The k-domination number γₖ(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ₂(T) ≥ γ₁(T)+1 and characterized extremal trees attaining this bound. In this paper we characterize all trees T with γ₂(T) = γ₁(T)+2.
Źródło:
Discussiones Mathematicae Graph Theory; 2011, 31, 4; 687-697
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 225

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