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Wyświetlanie 1-15 z 24
Tytuł:
Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree
Autorzy:
Henning, Michael A.
Marcon, Alister J.
Powiązania:
https://bibliotekanauki.pl/articles/31341173.pdf
Data publikacji:
2016-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
semitotal domination
trees
Opis:
Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γt2(G), is the minimum cardinality of a semitotal dominating set of G. We observe that γ(G) ≤ γt2(G) ≤ γt(G). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 1; 71-93
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Domination Subdivision and Domination Multisubdivision Numbers of Graphs
Autorzy:
Dettlaff, Magda
Raczek, Joanna
Topp, Jerzy
Powiązania:
https://bibliotekanauki.pl/articles/31343212.pdf
Data publikacji:
2019-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
domination subdivision number
domination multisubdivision number
trees
computational complexity
Opis:
The domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [10] that sd(T) ≤ 3 for any tree T. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number of a nonempty graph G as a minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. We show that msd(G) ≤ 3 for any graph G. The domination subdivision number and the domination multisubdivision number of a graph are incomparable in general, but we show that for trees these two parameters are equal. We also determine the domination multisubdivision number for some classes of graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 4; 829-839
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds On The Disjunctive Total Domination Number Of A Tree
Autorzy:
Henning, Michael A.
Naicker, Viroshan
Powiązania:
https://bibliotekanauki.pl/articles/31341124.pdf
Data publikacji:
2016-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
disjunctive total domination
trees
Opis:
Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $ \gamma_t(G) $. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance 2 from it. The disjunctive total domination number, $ \gamma_t^d (G) $, is the minimum cardinality of such a set. We observe that $ \gamma_t^d (G) \ge \gamma_t (G) $. A leaf of $G$ is a vertex of degree 1, while a support vertex of $G$ is a vertex adjacent to a leaf. We show that if $T$ is a tree of order $n$ with $ \mathcal{l} $ leaves and $s$ support vertices, then $ 2(n−\mathcal{l}+3) // 5 \le \gamma_t^d (T) \le (n+s−1)//2 $ and we characterize the families of trees which attain these bounds. For every tree $T$, we show have $ \gamma_t(T) // \gamma_t^d (T) <2 $ and this bound is asymptotically tight.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 1; 153-171
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On domination multisubdivision number of unicyclic graphs
Autorzy:
Raczek, J.
Powiązania:
https://bibliotekanauki.pl/articles/255095.pdf
Data publikacji:
2018
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
domination number
domination subdivision number
domination multisubdivision number
trees
unicyclic graphs
Opis:
The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems.
Źródło:
Opuscula Mathematica; 2018, 38, 3; 409-425
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total Domination Multisubdivision Number of a Graph
Autorzy:
Avella-Alaminos, Diana
Dettlaff, Magda
Lemańska, Magdalena
Zuazua, Rita
Powiązania:
https://bibliotekanauki.pl/articles/31339480.pdf
Data publikacji:
2015-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
(total) domination
(total) domination subdivision number
(total) domination multisubdivision number
trees
Opis:
The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdivision number is equal to the known total domination subdivision number. We also determine the total domination multisubdivision number for some classes of graphs and characterize trees T with msdγt (T) = 1.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 2; 315-327
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
Tytuł:
Domination Game Critical Graphs
Autorzy:
Bujtás, Csilla
Klavžar, Sandi
Košmrlj, Gašper
Powiązania:
https://bibliotekanauki.pl/articles/31234048.pdf
Data publikacji:
2015-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination number
domination game
domination game critical graphs
powers of cycles
trees
Opis:
The domination game is played on a graph $G$ by two players who alternately take turns by choosing a vertex such that in each turn at least one previously undominated vertex is dominated. The game is over when each vertex becomes dominated. One of the players, namely Dominator, wants to finish the game as soon as possible, while the other one wants to delay the end. The number of turns when Dominator starts the game on $G$ and both players play optimally is the graph invariant $ \gamma_g (G) $, named the game domination number. Here we study the $ \gamma_g$-critical graphs which are critical with respect to vertex predomination. Besides proving some general properties, we characterize $ \gamma_g$-critical graphs with $ \gamma_g = 2$ and with $ \gamma_g = 3$, moreover for each n we identify the (infinite) class of all $\gamma_g$-critical ones among the $n$th powers $ C_N^n$ of cycles. Along the way we determine $\gamma_{g} ( C_N^n ) $ for all $n$ and $N$. Results of a computer search for $ \gamma_g$-critical trees are presented and several problems and research directions are also listed.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 4; 781-796
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On locating and differentiating-total domination in trees
Autorzy:
Chellali, Mustapha
Powiązania:
https://bibliotekanauki.pl/articles/743043.pdf
Data publikacji:
2008
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
locating-total domination
differentiating-total domination
trees
Opis:
A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $γₜ^L(G)$ and $γₜ^D(G)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with l leaves and s support vertices, $γₜ^L(T) ≥ max{2(n+l-s+1)/5,(n+2-s)/2}$, and for a tree of order n ≥ 3, $γₜ^D(T) ≥ 3(n+l-s+1)/7$, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying $γₜ^L(T) = 2(n+l- s+1)/5$ or $γₜ^D(T) = 3(n+l-s+1)/7$.
Źródło:
Discussiones Mathematicae Graph Theory; 2008, 28, 3; 383-392
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:
Strong Equality Between the Roman Domination and Independent Roman Domination Numbers in Trees
Autorzy:
Chellali, Mustapha
Rad, Nader Jafari
Powiązania:
https://bibliotekanauki.pl/articles/30146596.pdf
Data publikacji:
2013-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Roman domination
independent Roman domination
strong equality
trees
Opis:
A Roman dominating function (RDF) on a graph $G = (V,E)$ is a function $ f : V \rightarrow {0, 1, 2} $ satisfying the condition that every vertex $ u $ for which $ f(u) = 0 $ is adjacent to at least one vertex $v$ for which $f(v) = 2$. The weight of an RDF is the value $ f(V (G)) = \Sigma_{u \in V (G) } f(u) $. An RDF $f$ in a graph $G$ is independent if no two vertices assigned positive values are adjacent. The Roman domination number $ \gamma_R (G) $ (respectively, the independent Roman domination number $ i_R(G) $) is the minimum weight of an RDF (respectively, independent RDF) on $G$. We say that $ \gamma_R(G)$ strongly equals $ i_R(G)$, denoted by $ \gamma_R (G) \equiv i_R(G)$, if every RDF on $G$ of minimum weight is independent. In this paper we provide a constructive characterization of trees $T$ with $ \gamma_R(T) \equiv i_R(T) $.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 2; 337-346
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Bounds on the 2-domination number in cactus graphs
Autorzy:
Chellali, M.
Powiązania:
https://bibliotekanauki.pl/articles/254915.pdf
Data publikacji:
2006
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
2-domination number
total domination number
independence number
cactus graphs
trees
Opis:
A 2-dominating set of a graph G is a set D of vertices of G such that every vertex not in S is dominated at least twice. The minimum cardinality of a 2-dominating set of G is the 2-domination number γ2(G). We show that if G is a nontrivial connected cactus graph with k(G) even cycles (k(G) ≥ 0), then γ2(G) ≥ γt(G) - k(G), and if G is a graph of order n with at most one cycle, then γ2(G) ≥ (n + l - s)/2 improving Fink and Jacobson's lower bound for trees with l > s, where γt(G), l and s are the total domination number, the number of leaves and support vertices of G, respectively. We also show that if T is a tree of order n ≥ 3, then γ2(T) ≤ β(T) + s - 1, where β(T) is the independence number of T.
Źródło:
Opuscula Mathematica; 2006, 26, 1; 5-12
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Independent Transversal Total Domination versus Total Domination in Trees
Autorzy:
Martínez, Abel Cabrera
Peterin, Iztok
Yero, Ismael G.
Powiązania:
https://bibliotekanauki.pl/articles/32083825.pdf
Data publikacji:
2021-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
independent transversal total domination number
total domination number
independence number
trees
Opis:
A subset of vertices in a graph G is a total dominating set if every vertex in G is adjacent to at least one vertex in this subset. The total domination number of G is the minimum cardinality of any total dominating set in G and is denoted by γt(G). A total dominating set of G having nonempty intersection with all the independent sets of maximum cardinality in G is an independent transversal total dominating set. The minimum cardinality of any independent transversal total dominating set is denoted by γtt(G). Based on the fact that for any tree T, γt(T) ≤ γtt(T) ≤ γt(T) + 1, in this work we give several relationships between γtt(T) and γt(T) for trees T which are leading to classify the trees which are satisfying the equality in these bounds.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 1; 213-224
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Constructive Characterization of Vertex Cover Roman Trees
Autorzy:
Martínez, Abel Cabrera
Kuziak, Dorota
Yero, Ismael G.
Powiązania:
https://bibliotekanauki.pl/articles/32083833.pdf
Data publikacji:
2021-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Roman domination
outer-independent Roman domination
vertex cover
vertex independence
trees
Opis:
A Roman dominating function on a graph G = (V(G), E(G)) is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The Roman dominating function f is an outer-independent Roman dominating function on G if the set of vertices labeled with zero under f is an independent set. The outer-independent Roman domination number γoiR(G) is the minimum weight w(f) = Σv∈V(G)f(v) of any outer-independent Roman dominating function f of G. A vertex cover of a graph G is a set of vertices that covers all the edges of G. The minimum cardinality of a vertex cover is denoted by α(G). A graph G is a vertex cover Roman graph if γoiR(G) = 2α(G). A constructive characterization of the vertex cover Roman trees is given in this article.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 1; 267-283
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A note on the p-domination number of trees
Autorzy:
Lu, Y.
Hou, X.
Xu, J.-M.
Powiązania:
https://bibliotekanauki.pl/articles/255183.pdf
Data publikacji:
2009
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
p-domination number
trees
Opis:
Let p be a positive integer and G = (V (G), E(G)) a graph. A p-dominating set of G is a subset S of V (G) such that every vertex not in S is dominated by at least p vertices in S. The p-domination number ϒp(G) is the minimum cardinality among the p-dominating sets of G. Let T be a tree with order n ≥ 2 and p ≥ 2 a positive integer. A vertex of V (T) is a p-leaf if it has degree at most p - 1, while a p-support vertex is a vertex of degree at least p adjacent to a p-leaf. In this note, we show that ϒp(T) ≥ (n + /Lp(T)/ - /Sp(T)/)/2, where Lp(T) and Sp(T) are the sets of p-leaves and p-support vertices of T, respectively. Moreover, we characterize all trees attaining this lower bound.
Źródło:
Opuscula Mathematica; 2009, 29, 2; 157-164
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the dominator colorings in trees
Autorzy:
Merouane, Houcine
Chellali, Mustapha
Powiązania:
https://bibliotekanauki.pl/articles/743280.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
dominator coloring
domination
trees
Opis:
In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different colors. A dominator coloring of a graph G is a proper coloring such that every vertex of V dominates all vertices of at least one color class (possibly its own class). The dominator chromatic number $χ_d(G)$ is the minimum number of color classes in a dominator coloring of G. Gera showed that every nontrivial tree T satisfies $γ(T)+1 ≤ χ_d(T) ≤ γ(T)+2$. In this note we characterize nontrivial trees T attaining each bound.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 4; 677-683
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 24

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