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Tytuł:
The irregularity of graphs under graph operations
Autorzy:
Abdo, Hosam
Dimitrov, Darko
Powiązania:
https://bibliotekanauki.pl/articles/30148232.pdf
Data publikacji:
2014-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
irregularity of graphs
total irregularity of graphs
graph operations
Zagreb indices
Opis:
The irregularity of a simple undirected graph $G$ was defined by Albertson [5] as $irr(G) = ∑_{uv∈E(G)} |dG(u) − dG(v)|$, where $d_G(u)$ denotes the degree of a vertex $u ∈ V (G)$. In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and symmetric difference. We give exact expressions or (sharp) upper bounds on the irregularity of graphs under the above mentioned operations
Źródło:
Discussiones Mathematicae Graph Theory; 2014, 34, 2; 263-278
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Grundy Total Domination Number in Product Graphs
Autorzy:
Brešar, Boštjan
Bujtás, Csilla
Gologranc, Tanja
Klavžar, Sandi
Košmrlj, Gašper
Marc, Tilen
Patkós, Balázs
Tuza, Zsolt
Vizer, Máté
Powiązania:
https://bibliotekanauki.pl/articles/32083828.pdf
Data publikacji:
2021-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total domination
Grundy total domination number
graph product
Opis:
A longest sequence $(v_1, . . ., v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, \(N(υ_i)\backslash\bigcup_{j=1}^{i-1}N(υ_j)≠∅\). The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $\gamma_{gr}^t(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $\gamma_{gr}^t(G×H)≥\gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express $\gamma_{gr}^t(G∘H)$ in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on $\gamma_{gr}^t(G⊠H)$ are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 1; 225-247
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total connected domination game
Autorzy:
Bujtás, Csilla
Henning, Michael A.
Iršič, Vesna
Klavžar, Sandi
Powiązania:
https://bibliotekanauki.pl/articles/2050904.pdf
Data publikacji:
2021
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
connected domination game
total connected domination game
graph product
tree
Opis:
The (total) connected domination game on a graph $G$ is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of $G$. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number $(\gamma_{tcg}(G))(\gamma_{cg(G)})$ of $G$. We show that $\gamma_{tcg}(G) \in \{\gamma_{cg}(G), \gamma_{cg}(G)+1, \gamma_{cg}(G)+2\}$, and consequently define $G$ as Class $i$ if $\gamma_{tcg}(G) = \gamma_{cg}(G)+i$ for $i \in \{0, 1, 2\}$. A large family of Class 0 graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minimum degree at least 2. We show that no tree is Class 2 and characterize Class 1 trees. We provide an infinite family of Class 2 bipartite graphs.
Źródło:
Opuscula Mathematica; 2021, 41, 4; 453-464
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Weak Total Resolvability In Graphs
Autorzy:
Casel, Katrin
Estrada-Moreno, Alejandro
Fernau, Henning
Rodríguez-Velázquez, Juan Alberto
Powiązania:
https://bibliotekanauki.pl/articles/31341108.pdf
Data publikacji:
2016-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
metric dimension
resolving set
weak total metric dimension
weak total resolving set
adjacency dimension
graph operations
Opis:
A vertex v ∈ V (G) is said to distinguish two vertices x, y ∈ V (G) of a graph G if the distance from v to x is di erent from the distance from v to y. A set W ⊆ V (G) is a total resolving set for a graph G if for every pair of vertices x, y ∈ V (G), there exists some vertex w ∈ W − {x, y} which distinguishes x and y, while W is a weak total resolving set if for every x ∈ V (G)−W and y ∈ W, there exists some w ∈ W −{y} which distinguishes x and y. A weak total resolving set of minimum cardinality is called a weak total metric basis of G and its cardinality the weak total metric dimension of G. Our main contributions are the following ones: (a) Graphs with small and large weak total metric bases are characterised. (b) We explore the (tight) relation to independent 2-domination. (c) We introduce a new graph parameter, called weak total adjacency dimension and present results that are analogous to those presented for weak total dimension. (d) For trees, we derive a characterisation of the weak total (adjacency) metric dimension. Also, exact figures for our parameters are presented for (generalised) fans and wheels. (e) We show that for Cartesian product graphs, the weak total (adjacency) metric dimension is usually pretty small. (f) The weak total (adjacency) dimension is studied for lexicographic products of graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 1; 185-210
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Intersection graph of gamma sets in the total graph
Autorzy:
Chelvam, T.
Asir, T.
Powiązania:
https://bibliotekanauki.pl/articles/743214.pdf
Data publikacji:
2012
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total graph
gamma sets
intersection graph
Hamiltonian
coloring
connectivity
domination number
Opis:
In this paper, we consider the intersection graph $I_{Γ}(ℤₙ)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{Γ}(ℤₙ)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{Γ}(ℤₙ)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{Γ}(ℤₙ)$.
Źródło:
Discussiones Mathematicae Graph Theory; 2012, 32, 2; 341-356
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Zig-zag facial total-coloring of plane graphs
Autorzy:
Czap, J.
Jendrol, S.
Voigt, M.
Powiązania:
https://bibliotekanauki.pl/articles/255827.pdf
Data publikacji:
2018
Wydawca:
Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:
plane graph
facial coloring total-coloring zig-zag coloring
Opis:
In this paper we introduce the concept of zig-zag facial total-coloring of plane graphs. We obtain lower and upper bounds for the minimum number of colors which is necessary for such a coloring. Moreover, we give several sharpness examples and formulate some open problems.
Źródło:
Opuscula Mathematica; 2018, 38, 6; 819-827
1232-9274
2300-6919
Pojawia się w:
Opuscula Mathematica
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Facial [r,s,t]-Colorings of Plane Graphs
Autorzy:
Czap, Július
Šugerek, Peter
Jendrol’, Stanislav
Valiska, Juraj
Powiązania:
https://bibliotekanauki.pl/articles/31343366.pdf
Data publikacji:
2019-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
plane graph
boundary walk
edge-coloring
vertex-coloring
total-coloring
Opis:
Let $G$ be a plane graph. Two edges are facially adjacent in $G$ if they are consecutive edges on the boundary walk of a face of $G$. Given nonnegative integers $r$, $s$, and $t$, a facial $[r, s, t]$-coloring of a plane graph $G = (V,E)$ is a mapping $f : V \cup E \rightarrow {1, . . ., k} $ such that $ |f(v_1) − f(v_2)| \ge r $ for every two adjacent vertices $ v_1 $ and $ v_2 $, $ | f(e_1) − f(e_2)| \ge s $ for every two facially adjacent edges $ e_1 $ and $ e_2 $, and $ | f(v) − f(e)| \ge t $ for all pairs of incident vertices $ v $ and edges $ e $. The facial $[r, s, t]$-chromatic number $ \overline{ \chi }_{r,s,t} (G) $ of $ G $ is defined to be the minimum $k$ such that $G$ admits a facial $[r, s, t]$-coloring with colors $1, . . ., k$. In this paper we show that $ \overline{ \chi }_{r,s,t} (G) \le 3r + 3s + t + 1 $ for every plane graph $G$. For some triplets $ [r, s, t] $ and for some families of plane graphs this bound is improved. Special attention is devoted to the cases when the parameters $r$, $s$, and $t$ are small.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 3; 629-645
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Twin Minus Total Domination Numbers In Directed Graphs
Autorzy:
Dehgardi, Nasrin
Atapour, Maryam
Powiązania:
https://bibliotekanauki.pl/articles/31341587.pdf
Data publikacji:
2017-11-27
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
twin minus total dominating function
twin minus total domination number
directed graph
Opis:
Let $ D = (V,A) $ be a finite simple directed graph (shortly, digraph). A function $ f : V \rightarrow {−1, 0, 1} $ is called a twin minus total dominating function (TMTDF) if $ f(N^−(v)) \ge 1 $ and $ f(N^+(v)) \ge 1 $ for each vertex $ v \in V $. The twin minus total domination number of $D$ is $\gamma_{mt}^\ast (D) = \text{min} \{ w(f) | f $ is a TMTDF of $ D \} $. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for $ \gamma_{mt}^\ast (D) $ in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 4; 989-1004
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Neighbor Product Distinguishing Total Colorings of Planar Graphs with Maximum Degree at least Ten
Autorzy:
Dong, Aijun
Li, Tong
Powiązania:
https://bibliotekanauki.pl/articles/32227944.pdf
Data publikacji:
2021-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total coloring
neighbor product distinguishing coloring
planar graph
Opis:
A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, . . ., k}. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) ≠ p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By X″∏(G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that Δ(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with Δ(G) ≥ 10. Moreover, for planar graph G with Δ(G) ≥ 11, it is neighbor product distinguishing (Δ(G) + 2)-total colorable, and the upper bound Δ(G) + 2 is tight.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 4; 981-999
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Note on the Total Detection Numbers of Cycles
Autorzy:
Escuadro, Henry E.
Fujie, Futaba
Musick, Chad E.
Powiązania:
https://bibliotekanauki.pl/articles/31339492.pdf
Data publikacji:
2015-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
vertex-distinguishing coloring
detectable labeling
detection number
total detection number
Hamiltonian graph
Opis:
Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . ., k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . ., ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of a detectable labeling c of a graph G is the sum of the labels assigned to the edges in G. The total detection number td(G) of G is defined by td(G) = min{val(c)}, where the minimum is taken over all detectable labelings c of G. We investigate the problem of determining the total detection numbers of cycles.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 2; 237-247
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
A Note on Total Graphs
Autorzy:
Forouhandeh, S.F.
Jafari Rad, N.
Vaqari Motlagh, B.H.
Patil, H.P.
Pandiya Raj, R.
Powiązania:
https://bibliotekanauki.pl/articles/31339326.pdf
Data publikacji:
2015-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total graph
central graph
middle graph
Mycielski graph
Opis:
Erratum Identification and corrections of the existing mistakes in the paper On the total graph of Mycielski graphs, central graphs and their covering numbers, Discuss. Math. Graph Theory 33 (2013) 361-371.
Źródło:
Discussiones Mathematicae Graph Theory; 2015, 35, 3; 585-587
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Fair Total Domination Number in Cactus Graphs
Autorzy:
Hajian, Majid
Rad, Nader Jafari
Powiązania:
https://bibliotekanauki.pl/articles/32083904.pdf
Data publikacji:
2021-05-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
fair total domination
cactus graph
Opis:
For k ≥ 1, a k-fair total dominating set (or just kFTD-set) in a graph G is a total dominating set S such that |N(v) ∩ S| = k for every vertex v ∈ V\S. The k-fair total domination number of G, denoted by ftdk(G), is the minimum cardinality of a kFTD-set. A fair total dominating set, abbreviated FTD-set, is a kFTD-set for some integer k ≥ 1. The fair total domination number of a nonempty graph G, denoted by ftd(G), of G is the minimum cardinality of an FTD-set in G. In this paper, we present upper bounds for the 1-fair total domination number of cactus graphs, and characterize cactus graphs achieving equality for the upper bounds.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 2; 647-664
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Partitioning a graph into a dominating set, a total dominating set, and something else
Autorzy:
Henning, Michael
Löwenstein, Christian
Rautenbach, Dieter
Powiązania:
https://bibliotekanauki.pl/articles/744065.pdf
Data publikacji:
2010
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
domination
total domination
domatic number
vertex partition
Petersen graph
Opis:
A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2010, 30, 4; 563-574
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Supermagic Graphs Having a Saturated Vertex
Autorzy:
Ivančo, Jaroslav
Polláková, Tatiana
Powiązania:
https://bibliotekanauki.pl/articles/30147220.pdf
Data publikacji:
2014-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
supermagic graph
saturated vertex
vertex-magic total labeling
Opis:
A graph is called supermagic if it admits a labeling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we establish some conditions for graphs with a saturated vertex to be supermagic. Inter alia we show that complete multipartite graphs K1,n,n and K1,2,...,2 are supermagic.
Źródło:
Discussiones Mathematicae Graph Theory; 2014, 34, 1; 75-84
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total edge irregularity strength of trees
Autorzy:
Ivančo, Jaroslav
Jendrol', Stanislav
Powiązania:
https://bibliotekanauki.pl/articles/743612.pdf
Data publikacji:
2006
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
graph labelling
tree
irregularity strength
total labellings
total edge irregularity strength
Opis:
A total edge-irregular k-labelling ξ:V(G)∪ E(G) → {1,2,...,k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree Δ on p vertices
tes(T) = max{⎡(p+1)/3⎤,⎡(Δ+1)/2⎤}.
Źródło:
Discussiones Mathematicae Graph Theory; 2006, 26, 3; 449-456
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
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