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### A bound on the $k$-domination number of a graph

Czechoslovak Mathematical Journal

Let $G$ be a graph with vertex set $V\left(G\right)$, and let $k\ge 1$ be an integer. A subset $D\subseteq V\left(G\right)$ is called a $k$-dominating set if every vertex $v\in V\left(G\right)-D$ has at least $k$ neighbors in $D$. The $k$-domination number ${\gamma }_{k}\left(G\right)$ of $G$ is the minimum cardinality of a $k$-dominating set in $G$. If $G$ is a graph with minimum degree $\delta \left(G\right)\ge k+1$, then we prove that ${\gamma }_{k+1}\left(G\right)\le \frac{|V\left(G\right)|+{\gamma }_{k}\left(G\right)}{2}.$ In addition, we present a characterization of a special class of graphs attaining equality in this inequality.

### A characterization of diameter-2-critical graphs with no antihole of length four

Open Mathematics

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence...

### A Characterization of Hypergraphs with Large Domination Number

Discussiones Mathematicae Graph Theory

Let H = (V, E) be a hypergraph with vertex set V and edge set E. A dominating set in H is a subset of vertices D ⊆ V such that for every vertex v ∈ V D there exists an edge e ∈ E for which v ∈ e and e ∩ D ≠ ∅. The domination number γ(H) is the minimum cardinality of a dominating set in H. It is known [Cs. Bujtás, M.A. Henning and Zs. Tuza, Transversals and domination in uniform hypergraphs, European J. Combin. 33 (2012) 62-71] that for k ≥ 5, if H is a hypergraph of order n and size m with all...

### A characterization of locating-total domination edge critical graphs

Discussiones Mathematicae Graph Theory

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, ${N}_{G}\left(u\right)\cap D\ne {N}_{G}\left(v\right)\cap D$. The locating-total domination number ${\gamma }_{L}^{t}\left(G\right)$ is the minimum cardinality of a locating-total dominating set...

### A characterization of roman trees

Discussiones Mathematicae Graph Theory

A Roman dominating function (RDF) on a graph G = (V,E) is a function f: V → 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 f is $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$. The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above by twice its domination number. Graphs which have Roman domination number equal to twice their domination number are called...

### A Characterization of Trees for a New Lower Bound on the K-Independence Number

Discussiones Mathematicae Graph Theory

Let k be a positive integer and G = (V,E) a graph of order n. A subset S of V is a k-independent set of G if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1. The maximum cardinality of a k-independent set of G is the k-independence number βk(G). In this paper, we show that for every graph [xxx], where χ(G), s(G) and Lv are the chromatic number, the number of supports vertices and the number of leaves neighbors of v, in the graph G, respectively. Moreover,...

### A characterization of (γₜ,γ₂)-trees

Discussiones Mathematicae Graph Theory

Let γₜ(G) and γ₂(G) be the total domination number and the 2-domination number of a graph G, respectively. It has been shown that: γₜ(T) ≤ γ₂(T) for any tree T. In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.

### A class of weakly perfect graphs

Czechoslovak Mathematical Journal

A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given.

### A Gallai-type equality for the total domination number of a graph

Discussiones Mathematicae Graph Theory

We prove the following Gallai-type equality γₜ(G) + εₜ(G) = p for any graph G with no isolated vertex, where p is the number of vertices of G, γₜ(G) is the total domination number of G, and εₜ(G) is the maximum integer s such that there exists a spanning forest F with s the number of pendant edges of F minus the number of star components of F.

### A Havel-Hakimi type procedure and a sufficient condition for a sequence to be potentially ${S}_{r,s}$-graphic

Czechoslovak Mathematical Journal

The split graph ${K}_{r}+\overline{{K}_{s}}$ on $r+s$ vertices is denoted by ${S}_{r,s}$. A non-increasing sequence $\pi =\left({d}_{1},{d}_{2},...,{d}_{n}\right)$ of nonnegative integers is said to be potentially ${S}_{r,s}$-graphic if there exists a realization of $\pi$ containing ${S}_{r,s}$ as a subgraph. In this paper, we obtain a Havel-Hakimi type procedure and a simple sufficient condition for $\pi$ to be potentially ${S}_{r,s}$-graphic. They are extensions of two theorems due to A. R. Rao (The clique number of a graph with given degree sequence, Graph Theory, Proc. Symp., Calcutta 1976, ISI Lect. Notes Series...

### A lower bound for the irredundance number of trees

Discussiones Mathematicae Graph Theory

Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska  presented in 2004 the sharp lower bound γ(T) ≥ (n(T) + 2 - n₁(T))/3. In this paper we prove ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of Lemańska's result. ...

### A lower bound on the independence number of a graph in terms of degrees

Discussiones Mathematicae Graph Theory

For a connected and non-complete graph, a new lower bound on its independence number is proved. It is shown that this bound is realizable by the well known efficient algorithm MIN.

### A maximum degree theorem for diameter-2-critical graphs

Open Mathematics

A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n 2/4⌋ and that the extremal graphs are the complete bipartite graphs K ⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n 0 where n 0 is a...

### A new bound on the domination number of graphs with minimum degree two.

The Electronic Journal of Combinatorics [electronic only]

### A New Characterization of Unichord-Free Graphs

Discussiones Mathematicae Graph Theory

Unichord-free graphs are defined as having no cycle with a unique chord. They have appeared in several papers recently and are also characterized by minimal separators always inducing edgeless subgraphs (in contrast to characterizing chordal graphs by minimal separators always inducing complete subgraphs). A new characterization of unichord-free graphs corresponds to a suitable reformulation of the standard simplicial vertex characterization of chordal graphs.

### A new lower bound on the density of vertex identifying codes for the infinite hexagonal grid.

The Electronic Journal of Combinatorics [electronic only]

### A new upper bound for the chromatic number of a graph

Discussiones Mathematicae Graph Theory

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ [(n+ω+1-α)/2]. Moreover, χ(G) ≤ [(n+ω-α)/2], if either ω + α = n + 1 and G is not a split graph or α + ω = n - 1 and G contains no induced ${K}_{\omega +3}-C₅$.

### A new upper bound on the total domination number of a graph.

The Electronic Journal of Combinatorics [electronic only]

### A not on dominating set with Maple.

The Journal of Nonlinear Sciences and its Applications

### A note on domination in bipartite graphs

Discussiones Mathematicae Graph Theory

DOMINATING SET remains NP-complete even when instances are restricted to bipartite graphs, however, in this case VERTEX COVER is solvable in polynomial time. Consequences to VECTOR DOMINATING SET as a generalization of both are discussed.

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