Retraction Note: Lower bound for the Sombor index of trees with a given total domination number

The total domination and total bondage numbers of extended de Bruijn and Kautz digraphs

Lower bounds for the domination number and the total domination number of direct product graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S. The total domination number, γt(G), of G is the minimum cardinality of a total dominating set of G. A set S of vertices in G is a disjunctive dominating set in G if every vertex not in S is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it in G. The disjunctive domination number, (G), of G is the minimum cardinality of a disjunctive dominating set in G. By definition, we have (T )≤γt (T ). In this paper, we provide a constructive characterization of the trees T achieving equality in this bound.

Total domination dot-stable graphs

Introduction: This paper presents an efficient algorithmic framework for computing the domination number (γ) and total domination number (γt) of graphs, with a focus on applications in collaboration networks. Using a greedy approach combined with minimality verification, the algorithm identifies dominating and total dominating sets that optimize connectivity within the graph structure. Objectives: To design and implement algorithms for computing the domination number of RNPCGs, focusing on both time complexity and practical efficiency. Methods: The methodology is demonstrated on the Rolf Nevanlinna Prize Collaboration Graph (RNPCG), showcasing its ability to handle complex, real-world datasets. Results reveal key insights into the connectivity and influence metrics of prominent academic networks, with implications for optimizing resource dissemination in such systems. The proposed algorithm balances computational efficiency with accuracy, offering a robust tool for graph-theoretic analysis in both theoretical and applied contexts Results: A detailed computational complexity analysis was conducted for both the domination number and total domination number algorithms. It was found that the proposed solutions offer polynomial-time complexity for graphs of moderate size, making them practical for typical use cases. However, the complexity does increase with the size of the graph, especially as the randomness in RNPCGs increases.The algorithms were also compared with exact brute-force solutions for small graphs, showing that our approach provides a significant reduction in computation time while maintaining accuracy. Conclusions: In this study, we developed an efficient algorithm for computing the domination number (γ) and total domination number (γt) in graphs, with a focus on its application to the Rolf Nevanlinna Prize Collaboration Graph (RNPCG). By leveraging a greedy approach and minimality verification, the algorithm effectively identified dominating and total dominating sets, providing insights into the connectivity and structural properties of the RNPCG. The analysis revealed the influence and connectivity of key vertices, illustrating the practical utility of domination metrics in understanding complex collaboration networks. The results emphasize the significance of domination parameters in optimizing resource dissemination and maintaining connectivity within academic or similar real-world networks.

Introduction: This paper presents an efficient algorithmic framework for computing the domination number (γ) and total domination number (γt) of graphs, with a focus on applications in collaboration networks. Using a greedy approach combined with minimality verification, the algorithm identifies dominating and total dominating sets that optimize connectivity within the graph structure. Objectives: To design and implement algorithms for computing the domination number of RNPCGs, focusing on both time complexity and practical efficiency. Methods: The methodology is demonstrated on the Rolf Nevanlinna Prize Collaboration Graph (RNPCG), showcasing its ability to handle complex, real-world datasets. Results reveal key insights into the connectivity and influence metrics of prominent academic networks, with implications for optimizing resource dissemination in such systems. The proposed algorithm balances computational efficiency with accuracy, offering a robust tool for graph-theoretic analysis in both theoretical and applied contexts Results: A detailed computational complexity analysis was conducted for both the domination number and total domination number algorithms. It was found that the proposed solutions offer polynomial-time complexity for graphs of moderate size, making them practical for typical use cases. However, the complexity does increase with the size of the graph, especially as the randomness in RNPCGs increases.The algorithms were also compared with exact brute-force solutions for small graphs, showing that our approach provides a significant reduction in computation time while maintaining accuracy. Conclusions: In this study, we developed an efficient algorithm for computing the domination number (γ) and total domination number (γt) in graphs, with a focus on its application to the Rolf Nevanlinna Prize Collaboration Graph (RNPCG). By leveraging a greedy approach and minimality verification, the algorithm effectively identified dominating and total dominating sets, providing insights into the connectivity and structural properties of the RNPCG. The analysis revealed the influence and connectivity of key vertices, illustrating the practical utility of domination metrics in understanding complex collaboration networks. The results emphasize the significance of domination parameters in optimizing resource dissemination and maintaining connectivity within academic or similar real-world networks.

Let G = ( V,E ) be a graph without an isolated vertex. A set S ⊆ V is a total dominating set if S is a dominating set, and the induced subgraph G [ S ] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G . A set D ⊆ V is a total outer-connected dominating set if D is a total dominating set, and the induced subgraph G [ V − D ] is connected. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G . In this paper we generalize the total outer-connected domination number in graphs. Let k ≥ 1 be an integer. A set D ⊆ V is a total outer- k -connected component dominating set if D is a total dominating and the induced subgraph G [ V − D ] has exactly k connected component(s). The total outer- k -connected component domination number of G , denoted by γ k tc ( G ) , is the minimum cardinality of a total outer- k -connected component dominating set of G . We obtain several general results and bounds for γ k tc ( G ), and we determine exact values of γ k tc ( G ) for some special classes of graphs G .

On graphs for which the connected domination number is at most the total domination number

Total domination and matching numbers in graphs with all vertices in triangles

For a given graph G without isolated vertex we consider a function f: V(G) rightarrow {0,1,2}. For every iin {0,1,2}, let V_i={vin V(G):; f(v)=i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V_0 is adjacent to at least one vertex in V_2; (ii) V_0 is an independent set; and (iii) the subgraph induced by V_1cup V_2 has no isolated vertex. The minimum possible weight omega (f)=sum _{vin V(G)}f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G is a hop dominating set of G if every vertex outside S is 2-step dominated by some vertex of S. The hop domination number, $$\gamma _{h}(G)$$, of G is the minimum cardinality of a hop dominating set of G. A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to at least one vertex of S. The total domination number, $$\gamma _{t}(G)$$, of G is the minimum cardinality of a total dominating set of G. It is known that if G is a triangle-free graph, then $$\gamma _{h}(G)\le \gamma _{t}(G)$$. But there are connected graphs G for which the difference $$\gamma _{h}(G)-\gamma _{t}(G)$$ can be made arbitrarily large. It would be interesting to find other classes of graphs G that satisfy $$\gamma _{h}(G)\le \gamma _{t}(G)$$. In this paper, we study the relationship between total domination number and hop domination number in diamond-free graph. We prove that if G is diamond-free graph of order n with the exception of two special graphs, then $$\gamma _{h}(G)- \gamma _{t}(G)\le \frac{n}{6}$$. Furthermore, we find two subclasses of diamond-free graphs G that satisfy $$\gamma _{h}(G)\le \gamma _{t}(G)$$ and generalize the known result.

Let $$G=(V,E)$$G=(V,E) be a simple graph without isolated vertices. A set $$S$$S of vertices is a total dominating set of a graph $$G$$G if every vertex of $$G$$G is adjacent to some vertex in $$S$$S. A paired dominating set of $$G$$G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369---378, 2014) computed the exact values of total and paired domination numbers of Cartesian product $$C_n\square C_m$$CnźCm for $$m=3,4$$m=3,4. Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369---378, 2014) to graph bundle and compute the total domination number and the paired domination number of $$C_m$$Cm bundles over a cycle $$C_n$$Cn for $$m=3,4$$m=3,4. Moreover, we give the exact value for the total domination number of Cartesian product $$C_n\square C_5$$CnźC5 and some upper bounds of $$C_m$$Cm bundles over a cycle $$C_n$$Cn where $$m\ge 5$$mź5.

A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The total domination number, $$\gamma _t(G)$$, is the minimum cardinality of a total dominating set of G. A cactus is a connected graph in which every edge belongs to at most one cycle. Equivalently, a cactus is a connected graph in which every block is an edge or a cycle. Let G be a connected graph of order $$n \ge 2$$ with $$k \ge 0$$ cycles and $$\ell $$ leaves. Recently, the authors have proved that $$\gamma _t(G) \ge \frac{1}{2}(n-\ell +2) - k$$. As a consequence of this bound, $$\gamma _t(G) = \frac{1}{2}(n-\ell +2+m) - k$$ for some integer $$m \ge 0$$. In this paper, we characterize the class of cactus graphs achieving equality in this bound, thereby providing a classification of all cactus graphs according to their total domination number.

Total domination in planar graphs of diameter two

A total dominator coloring of a graph $$G$$G is a proper coloring of the vertices of $$G$$G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $$\chi _d^t(G)$$?dt(G) of $$G$$G is the minimum number of colors among all total dominator coloring of $$G$$G. A total dominating set of $$G$$G is a set $$S$$S of vertices such that every vertex in $$G$$G is adjacent to at least one vertex in $$S$$S. The total domination number $$\gamma _t(G)$$?t(G) of $$G$$G is the minimum cardinality of a total dominating set of $$G$$G. We establish lower and upper bounds on the total dominator chromatic number of a graph in terms of its total domination number. In particular, we show that every graph $$G$$G with no isolated vertex satisfies $$\gamma _t(G) \le \chi _d^t(G) \le \gamma _t(G) + \chi (G)$$?t(G)≤?dt(G)≤?t(G)+?(G), where $$\chi (G)$$?(G) denotes the chromatic number of $$G$$G. We establish properties of total dominator colorings in trees. We characterize the trees $$T$$T for which $$\gamma _t(T) = \chi _d^t(T)$$?t(T)=?dt(T). We prove that if $$T$$T is a tree of $$n \ge 2$$n?2 vertices, then $$\chi _d^t(T) \le 2(n+1)/3$$?dt(T)≤2(n+1)/3 and we characterize the trees achieving equality in this bound.