Let G = (V, E) be a graph and k >= 1. A subset S of V is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. We denote minimum cardinality of a k-path vertex cover in G by psi(k)(G) and call it the k-path vertex cover number of G. A set D subset of V of vertices of G is said to be a distance k-dominating set of G if the distance between each vertex u is an element of V \ D and D is at most k. In the paper we study a relationship between the k-path vertex cover and the k-distance domination number of trees. Moreover, we present a full characterization of trees such that each vertex belongs to some minimum k-path vertex cover.

We investigate a proper arc colouring of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals of integers. Oriented graphs having such colourings are called interval colourable. We analyse the parameter icr(D), which denotes the minimum number of arcs of D that should be reversed so that a resulting oriented graph is interval colourable. We prove that for each non-negative integer p there exists an oriented graph D with the property p≤icr(D)≤4p+1. We show that icr(D) is not monotone with respect to taking subdigraphs. We give an upper bound on icr(D) if D is an arbitrary oriented graph with finite parameter icr(D), and next if D is any orientation of a 2-degenerate graph. Also the exact values of icr(D) for all orientations D of some generalized Hertz graphs and generalized Sevastjanov rosettes are given. Based on special results concerning the still open problem of finding the largest possible transitive subtournament in a tournament (posed by Erdős and Moser in 1964) we refine the upper bound on icr(D) if D is a tournament.

We consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r -regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

En este art́ıculo consideramos un concepto de coloraciones en gráficas, las coloraciones consecutivas en aristas, y lo extendemos a la definición de coloraciones consecutivas de flechas en digráficas. Una coloración propia por aristas de G se dice consecutiva si para todo vértice v ∈ V (G), el conjunto de colores de las aristas incidentes al vértice v ∈ V (G) forman un intervalo de enteros. Si una gráfica G tiene una coloración consecutiva decimos que G es consecutivamente coloreable. El concepto de gráfica continuamente coloreable (o coloreable por intervalos), fue definido en 1987 por Asratian y Kamalian [1] y después (llamado coloraciones consecutivas) estudiado por K. Giaro, M. Kubale y M. Ma lafiejski [8]. Es importante notar que no todas las gráficas son consecutivamente coloreables, de hecho el ejemplo de gráfica de orden más pequeño que no es consecutivamente coloreable es el ciclo C3. Una coloración consecutiva de una digráfica D es aquella en la que para todo vértice v ∈ V (D) se cumple que al colorear todas las flechas que inician en v esta coloración forma un intervalo de enteros y al colorear todas las flechas que terminan en v tal coloración forma un intervalo de enteros. En la sección 2 damos la motivación del estudio de coloraciones consecutivas y algunos conceptos básicos que nos servirán de apoyo para el

The minimum number of total independent partition sets of V boolean OR E of a graph G = (V, E) is called the total chromatic number of G, denoted by chi '' (G). If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V boolean OR E is called the equitable total chromatic number, and is denoted by chi '' = (G). In this paper we consider equitable total coloring of coronas of cubic graphs, G circle H. It turns out that independently on the values of equitable total chromatic number of factors G and H, equitable total chromatic number of corona G circle H is equal to Delta(G circle H) + 1. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.

A graph G for which γR(G)=2γ(G) is the Roman graph, and if γRwc(G)=2γwc(G), then G is the weakly connected Roman graph. In this paper, we show that the decision problem of whether a bipartite graph is Roman is a co-NP-hard problem. Next, we prove similar results for weakly connected Roman graphs. We also study Roman trees improving the result of M.A. Henning’s A characterization of Roman trees, Discuss. Math. Graph Theory 22 (2002). Moreover, we give a characterization of weakly connected Roman trees.

An adjacent vertex distinguishing total k-coloring f of a graph G is a proper total k-coloring of G such that no pair of adjacent vertices has the same color sets. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph G has adjacent vertex distinguishing total (∆(G) + 3)-coloring. In this paper we confirm the conjecture for many coronas, in particular for generalized, simple and l-coronas of graphs, not relating the results to particular graph classes.

Let e be an edge of a connected simple graph G. The graph obtained by removing (resp. subdividing) an edge e from G is denoted by G−e (resp. Ge). As usual, γ(G) denotes the domination number of G. We call G an SR-graph if for every edge e of G, γ(G−e)=γ(Ge); and G is an ASR-graph if for every edge e of G, γ(G−e)≠γ(Ge). In this work we give several examples of SR and ASR-graphs. We characterize SR-trees and show that ASR-graphs are efficient and γsd-critical. Consequently, ASR graphs are γ-insensitive and satisfy Vizing’s Conjecture.

The neighbourhood of a vertex v of a graph G is the set N(v) of all vertices adjacent to v in G. For D⊆V(G) we define D¯=V(G)∖D. A set D⊆V(G) is called a super dominating set if for every vertex u∈D¯, there exists v∈D such that N(v)∩D¯={u}. The super domination number of G is the minimum cardinality among all super dominating sets in G. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard.

In this paper, we study the concept of convex domination in maximal outerplanar graphs. For this class of graphs, we discuss several properties of this domination parameter, in particular, we provide upper bounds on the convex domination number and study effects on the convex domination number when a maximal outerplanar graph is modified by flipping a diagonal. We also propose a linear time algorithm for computing a minimum convex dominating set in a given maximal outerplanar graph. In addition, we consider the concept of convex guard sets in maximal outerplanar graphs and simple polygons.

A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(\gamma_t-\tau)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(\gamma_t-\tau)$-set.

Let $D=(V,A)$ be a digraph. A subset $S$ of $V$ is called a twin dominating set of $D$ if for every vertex $v\in V-S$, there exists vertices $u_1,u_2 \in S$ such that $(v,u_1)$ and $(u_2,v)$ are arcs in $D$. The minimum cardinality of a twin dominating set in $D$ is called the twin domination number of $D$ and is denoted by $\gamma ^{*}(D)$. In \cite{ChDSS}, is defined the concept of upper orientable twin domination number of a graph $G$, $DOM^{*}(G)=max\{ \gamma ^{*}(D)|D \ \text{is an orientation of G} \}.$ In \cite{AES}, it is conjectured that for the complete graph $K_n$ with $n\geq 8$, $DOM^{*}(K_n)=\left\lceil \frac{n+1}{2}\right\rceil$. In this work we prove that $DOM^{*}(K_n)\leq\left\lceil \frac{n}{2}\right\rceil$ for all even number $n\geq 8$.

Let D=(V,A) be a digraph. A subset S of V is called a twin dominating set of D if for every vertex v∈ V-S, there exists vertices u_1,u_2 ∈ S such that (v,u_1) and (u_2,v) are arcs in D. The minimum cardinality of a twin dominating set in D is called the twin domination number of D and is denoted by γ ^*(D). The upper orientable twin domination number of a graph G is DOM^*(G)=max{γ ^*(D)|D is an orientation of G}. It has been conjectured that for the complete graph K_n with n≥ 8, DOM^*(K_n)=⌈n+1/2⌉. In this work we prove DOM^*(K_8)= DOM^*(K_9)= 4 and establish new upper bounds for DOM^*(K_n), disproving the same above conjecture for all n ≥ 8.

In this note we present an alternative proof of the result by Dorfling et al. (Discrete Math 339(3):1180–1188, 2016) establishing that any maximal outerplanar graph of order \(n \ge 5\) has a total dominating set of size at most \(\lfloor \frac{2n}{5}\rfloor \), apart from two exceptions. In addition, we briefly discuss a relation between total domination in maximal outerplanar graphs and the concept of watched guards in simple polygons.

A non-isolated vertex x is an element of V(G) is called C-3-free if x belongs to no triangle of G. In [1] Burger, Mynhardt and Weakley introduced the idea of universal fixers. Let G = (V, E) be a graph with n vertices and G' a copy of G. For a bijective function pi : V(G) -> V(G'), we define the prism pi G of G as follows: V (pi G) = V (G) boolean OR V (G') and E(pi G) = E(G) boolean OR E(G') boolean OR where M-pi = {u pi(u) : u is an element of V(G)}. Let gamma(G) be the domination number of G. If gamma(pi G) = gamma(G) for any bijective function pi, then G is called a universal fixer. In [3] it is conjectured that the only universal fixer is the edgeless graph (K) over bar (n). In this note, we prove that any graph G with C-3-free vertices is not a universal fixer graph.

We consider (psi(k) - gamma(k-1))-perfect graphs, i.e., graphs G for which psi(k)(H) = gamma(k-1)(H) for any induced subgraph H of G, where psi(k) and gamma(k-1) are the k-path vertex cover number and the distance (k - 1) -domination number, respectively. We study (psi(k) - gamma(k-1))-perfect paths, cycles and complete graphs for k >= 2. Moreover, we provide a complete characterisation of (psi(2) - gamma(1))-perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging to this family.

Abstract A set of vertices D of a graph G is a distance 2-dominating set of G if the distance between each vertex u ∊ (V (G) − D) and D is at most two. Let γ2(G) denote the size of a smallest distance 2-dominating set of G. For any permutation π of the vertex set of G, the prism of G with respect to π is the graph πG obtained from G and a copy G′ of G by joining u ∊ V(G) with v′ ∊ V(G′) if and only if v′ = π(u). If γ2(πG) = γ2(G) for any permutation π of V(G), then G is called a universal γ2-fixer. In this work we characterize the cycles and paths that are universal γ2-fixers.

In this paper some results on the super domination number are obtained. We prove that if T is a tree with at least three vertices, then \(\frac{n}{2}\le \gamma _{sp}(T)\le n-s\), where s is the number of support vertices in T and we characterize the extremal trees.

In 1978, C. Thomassen proved that in any graph one can destroy all the longest cycles by deleting at most one third of the vertices. We show that for graphs with circumference k≤8 it suffices to remove at most 1/k of the vertices. The Petersen graph demonstrates that this result cannot be extended to include k=9 but we show that in every graph with circumference nine we can destroy all 9-cycles by removing 1/5 of the vertices. We consider the analogous problem for digraphs and show that for digraphs with circumference k=2,3, it suffices to remove 1/k of the vertices. However this does not hold for k≥4.

Let T be a 3-partite tournament and F3(T) be the set of vertices of T not in triangles. We prove that, if the global irregularity of T, ig(T), is one and |F3(T)|>3, then F3(T) must be contained in one of the partite sets of T and |F3(T)|≤⌊k+14⌋+1, which implies |F3(T)|≤⌊n+512⌋+1, where k is the size of the largest partite set and n the number of vertices of T. Moreover, we give some upper bounds on the number, as well as results on the structure of said vertices within the digraph, depending on its global irregularity.