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### On well-dominated direct, Cartesian and strong product graphs [article]

Douglas F. Rall
2021 arXiv   pre-print
If each minimal dominating set in a graph is a minimum dominating set, then the graph is called well-dominated.  ...  In this paper we give a complete characterization of nontrivial direct products that are well-dominated.  ...  Let G be a nontrivial, connected graph and suppose that D is a minimum dominating set of G with a vertex u ∈ c(D).  ...

### Restrained weakly connected independent domination in the join of graphs

Rene E. Leonida
2015 Applied Mathematical Sciences
A connected graph is constructed with a given weakly connected independent domination number, restrained weakly connected independent domination number, and maximum weakly connected independent domination  ...  In this paper, we explore the concept of restrained weakly connected independent domination in graphs.  ...  Hence, G = C ∪ {v} , where C is a nontrivial component of G with γ( C ) = 1. Suppose G = C ∪ {v} , where C is a nontrivial component of G with γ( C ) = 1. Then i rw (K 1 + G) = 1.  ...

### A simple upper bound for the hamiltonian index of a graph

Marko Lovrečič Saražin
1994 Discrete Mathematics
Let G be a connected graph other than a path and ham(G), A (G) be its hamiltonian index and maximal degree, respectively. It is proved that ham(G)~<] V(G)J--A(G).  ...  If B(G) possesses a dominating circuit D', then G need not contain one even if D' goes through all 'nontrivial' vertices, i.e. nontrivial 3-components of G.  ...  If deg,(u)= 1 and e is the only edge incident with U, then e is a vertex in L(G) which is either a trivial 3-component or contained in a nontrivial 3-component H'.  ...

### Restrained weakly connected domination in the join and corona of graphs

Rene E. Leonida
2015 Applied Mathematical Sciences
A connected graph is constructed with a given order, weakly connected domination number, and restrained weakly connected domination number. Mathematics Subject Classification: 05C69  ...  In this paper, we explore the concept of restrained weakly connected domination in graphs.  ...  Clearly, S is a weakly connected dominating set of K 1 + G. Let x ∈ V (K 1 + G)\S. Then x ∈ V (G)\{u}. Thus, x ∈ V ( C ) for some nontrivial component C of G.  ...

### Locating-total dominating sets in twin-free graphs: a conjecture [article]

Florent Foucaud, Michael A. Henning
2016 arXiv   pre-print
We conjecture that if G is a twin-free graph of order n with no isolated vertex, then LT(G) ≤2/3n. We prove the conjecture for graphs without 4-cycles as a subgraph.  ...  A locating-total dominating set of G is a total dominating set D of G with the additional property that every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices  ...  We remark that there are (twin-free) graphs with total domination number two and arbitrarily large location-total domination number.  ...

### Triple connected complementary tree domination number of a graph

2013 International Mathematical Forum
A subset S of V of a nontrivial graph connected graph G is said to be a complementary tree dominating set, if S is a dominating set and the induced sub graph is a tree.  ...  A subset S of V of a nontrivial connected graph G is said to be triple connected dominating set, if S is a dominating set and the induced sub graph is triple connected.  ...  A maximal connected subgraph of a graph G is called a component of G. The number of components of G is denoted by (G).  ...

### On Chordal-k-Generalized Split Graphs [article]

Andreas Brandstädt, Raffaele Mosca
2017 arXiv   pre-print
A graph G is a chordal-k-generalized split graph if G is chordal and there is a clique Q in G such that every connected component in G[V ∖ Q] has at most k vertices.  ...  Moreover, we characterize a very special case of chordal-2-generalized split graphs for which the Efficient Domination problem is -complete.  ...  Let Q be a clique in G with smallest 2-nontrivial component Z Q of G[V \ Q] with respect to all cliques in G. Clearly, Z Q is also the nontrivial component of G[V \ Q 1 ], i.e., Z Q 1 = Z Q .  ...

### Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

Hajo Broersma, Fedor V. Fomin, Pim van 't Hof, Daniël Paulusma
2011 Algorithmica
For a claw-free graph G, finding a longest cycle is equivalent to finding a closed trail (i.e., a connected even subgraph, possibly consisting of a single vertex) that dominates the largest number of edges  ...  of some associated graph H.  ...  An optimum nontrivial closed trail or ONCT of H is a nontrivial closed trail of H that dominates at least as many edges of H as any other nontrivial closed trail of H.  ...

### On Restrained Strong Resolving Domination in Graphs

Helyn Cosinas Sumaoy, Helen M. Rara
2021 European Journal of Pure and Applied Mathematics
strong resolving domination number of each of these graphs.  ...  In this paper, we present characterizations of the restrained strong resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the exact value of the restrained  ...  Let G be a nontrivial connected graph of order n with γ(G) = 1 and K 1 = ⟨v⟩.  ...

### Nonsplit Roman Domination In Graphs

C.Sivagnanam
2016 Zenodo
The graph G = (V,E) we mean a finite, undirected, connected graph with neither loops nor multiple edges. The order and size of G are denoted by n and m respectively.  ...  Then γ ns = γ nsr (G) if and only if G is a trivial graph. ¾ Theorem 2.17 Let G be a nontrivial graph of order n.  ...  Proof Let v ∈ V (G) such that N (v) has a component of order n − γ ns (G). Let G 1 be the component of N (v) with |V (G 1 )| = n − γ ns (G).  ...

### On dominator colorings in graphs

S ARUMUGAM, JAY BAGGA, K RAJA CHANDRASEKAR
2012 Proceedings of the Indian Academy of sciences. Mathematical sciences
In this paper we present several results on graphs with χ d (G) = χ(G) and χ d (G) = γ (G) where χ(G) and γ (G) denote respectively the chromatic number and the domination number of a graph G.  ...  A dominator coloring of a graph G is a proper coloring of G in which every vertex dominates every vertex of at least one color class.  ...  Clearly V − V 1 is a nontrivial bipartite graph with bipartition C 1 , C 2 . Now,suppose V 1 is not a dominating set.  ...

### On the independence number of traceable 2-connected claw-free graphs

Shipeng Wang, Liming Xiong
2018 Discussiones Mathematicae Graph Theory
As a corollary, we also show that every 2-connected claw-free graph with independence number α(G) ≤ 5 is traceable.  ...  In this article, we show that every 2-connected claw-free graph with independence number α(G) ≤ 6 is traceable or belongs to two exceptional families of welldefined graphs.  ...  Every nontrivial component of G − V (C) has a dominating path P such that one of the end vertices of P is adjacent to C. Proof.  ...

### Edgeless graphs are the only universal fixers [article]

Kirsti Wash
2013 arXiv   pre-print
G is said to be a universal fixer if the domination number of π G is equal to the domination number of G for all π of V(G).  ...  Given two disjoint copies of a graph G, denoted G^1 and G^2, and a permutation π of V(G), the graph π G is constructed by joining u ∈ V(G^1) to π(u) ∈ V(G^2) for all u ∈ V(G^1).  ...  This observation along with the results of Mynhardt and Xu [5] allow us to consider only nontrivial connected graphs with domination number at least 4.  ...

### On well-edge-dominated graphs [article]

Sarah E. Anderson and Kirsti Kuenzel and Douglas F. Rall
2021 arXiv   pre-print
A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum.  ...  We also characterize the well-edge-dominated split graphs and Cartesian products.  ...  The set consisting of all the vertices that are incident with at least one edge in a minimum edge dominating set is a vertex dominating set in a nontrivial connected graph.  ...

### The st-bond polytope on series-parallel graphs

Roland Grappe, Mathieu Lacroix
2018 Reserche operationelle
We also show that the st-bond polytope is the intersection of the st-cut dominant and the bond polytope. Mathematics Subject Classification. 90C27, 90C35 and 90C57  ...  In this paper, we provide a linear description of the st-bond polytope on series-parallel graphs.  ...  Hence, partitioning a smart grid into two such areas reduces to finding an st-bond with additional properties.  ...
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