9,014 research outputs found

    A new approach on locally checkable problems

    Full text link
    By providing a new framework, we extend previous results on locally checkable problems in bounded treewidth graphs. As a consequence, we show how to solve, in polynomial time for bounded treewidth graphs, double Roman domination and Grundy domination, among other problems for which no such algorithm was previously known. Moreover, by proving that fixed powers of bounded degree and bounded treewidth graphs are also bounded degree and bounded treewidth graphs, we can enlarge the family of problems that can be solved in polynomial time for these graph classes, including distance coloring problems and distance domination problems (for bounded distances)

    A note on the double Roman domination number of graphs

    Get PDF
    summary:For a graph G=(V,E)G=(V,E), a double Roman dominating function is a function f ⁣:V{0,1,2,3}f\colon V\rightarrow \{0,1,2,3\} having the property that if f(v)=0f(v)=0, then the vertex vv must have at least two neighbors assigned 22 under ff or one neighbor with f(w)=3f(w)=3, and if f(v)=1f(v)=1, then the vertex vv must have at least one neighbor with f(w)2f(w)\geq 2. The weight of a double Roman dominating function ff is the sum f(V)=vVf(v)f(V)=\sum \nolimits _{v\in V}f(v). The minimum weight of a double Roman dominating function on GG is called the double Roman domination number of GG and is denoted by γdR(G)\gamma _{\rm dR}(G). In this paper, we establish a new upper bound on the double Roman domination number of graphs. We prove that every connected graph GG with minimum degree at least two and GC5G\neq C_{5} satisfies the inequality γdR(G)1311n\gamma _{\rm dR}(G)\leq \lfloor \frac {13}{11}n\rfloor . One open question posed by R. A. Beeler et al. has been settled

    Signed double Roman domination on cubic graphs

    Full text link
    The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from {±1,2,3}\{\pm{}1,2,3\} to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility is given whenever (i) vertices labeled ±1\pm{}1 have at least one neighbor with label in {2,3}\{2,3\}; (ii) each vertex labeled 1-1 has one 33-labeled neighbor or at least two 22-labeled neighbors; and (iii) the sum of labels over the closed neighborhood of any vertex is positive. The cumulative weight of an optimal labeling is called signed double Roman domination number (SDRDN). In this work, we first consider the problem on general cubic graphs of order nn for which we present a sharp n/2+Θ(1)n/2+\Theta(1) lower bound for the SDRDN by means of the discharging method. Moreover, we derive a new best upper bound. Observing that we are often able to minimize the SDRDN over the class of cubic graphs of a fixed order, we then study in this context generalized Petersen graphs for independent interest, for which we propose a constraint programming guided proof. We then use these insights to determine the SDRDNs of subcubic 2×m2\times m grid graphs, among other results

    Double Roman domination and domatic numbers of graphs

    Get PDF
    A double Roman dominating function on a graph GG with vertex set V(G)V(G) is defined in \cite{bhh} as a function‎ ‎f:V(G){0,1,2,3}f:V(G)\rightarrow\{0,1,2,3\} having the property that if f(v)=0f(v)=0‎, ‎then the vertex vv must have at least two‎ ‎neighbors assigned 2 under ff or one neighbor ww with f(w)=3f(w)=3‎, ‎and if f(v)=1f(v)=1‎, ‎then the vertex vv must have‎ ‎at least one neighbor uu with f(u)2f(u)\ge 2‎. ‎The weight of a double Roman dominating function ff is the sum‎ ‎vV(G)f(v)\sum_{v\in V(G)}f(v)‎, ‎and the minimum weight of a double Roman dominating function on GG is the double Roman‎ ‎domination number γdR(G)\gamma_{dR}(G) of GG‎. ‎A set {f1,f2,,fd}\{f_1,f_2,\ldots,f_d\} of distinct double Roman dominating functions on GG with the property that‎ ‎i=1dfi(v)3\sum_{i=1}^df_i(v)\le 3 for each vV(G)v\in V(G) is called in \cite{v} a double Roman dominating family (of functions)‎ ‎on GG‎. ‎The maximum number of functions in a double Roman dominating family on GG is the double Roman domatic number‎ ‎of GG‎. ‎In this note we continue the study of the double Roman domination and domatic numbers‎. ‎In particular‎, ‎we present‎ ‎a sharp lower bound on γdR(G)\gamma_{dR}(G)‎, ‎and we determine the double Roman domination and domatic numbers of some‎ ‎classes of graphs

    Advances in Discrete Applied Mathematics and Graph Theory

    Get PDF
    The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

    Upper bounds for covering total double Roman domination

    Get PDF
    Let G = (V, E) be a finite simple graph where V = V (G) and E = E(G). Suppose that G has no isolated vertex. A covering total double Roman dominating function (CT DRD function) f of G is a total double Roman dominating function (T DRD function) of G for which the set {v ∈ V (G)|f(v) ≠ 0} is a covering set. The covering total double Roman domination number γctdR(G) is the minimum weight of a CT DRD function on G. In this work, we present some contributions to the study of γctdR(G)-function of graphs. For the non star trees T, we show that γctdR(T) ≤ 4n(T )+5s(T )−4l(T )/3, where n(T), s(T) and l(T) are the order, the number of support vertices and the number of leaves of T respectively. Moreover, we characterize trees T achieve this bound. Then we study the upper bound of the 2-edge connected graphs and show that, for a 2-edge connected graphs G, γctdR(G) ≤ 4n/3 and finally, we show that, for a simple graph G of order n with δ(G) ≥ 2, γctdR(G) ≤ 4n/3 and this bound is sharp.Publisher's Versio

    Domination parameters with number 2: Interrelations and algorithmic consequences

    Get PDF
    In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γw2(G), the 2-domination number, γ2(G), the {2}-domination number, γ{2}(G), the double domination number, γ×2(G), the total {2}-domination number, γt{2}(G), and the total double domination number, γt×2(G), where G is a graph in which the corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G), and two classical parameters, the domination number, γ(G), and the total domination number, γt(G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Brešar, Boštjan. Institute of Mathematics, Physics and Mechanics; Eslovenia. University of Maribor; EsloveniaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Milanič, Martin. University of Primorska; EsloveniaFil: Safe, Martin Dario. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin

    Domination parameters with number 2: interrelations and algorithmic consequences

    Full text link
    In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 22-domination number, γw2(G)\gamma_{w2}(G), the 22-domination number, γ2(G)\gamma_2(G), the {2}\{2\}-domination number, γ{2}(G)\gamma_{\{2\}}(G), the double domination number, γ×2(G)\gamma_{\times 2}(G), the total {2}\{2\}-domination number, γt{2}(G)\gamma_{t\{2\}}(G), and the total double domination number, γt×2(G)\gamma_{t\times 2}(G), where GG is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G)\gamma_R(G), and two classical parameters, the domination number, γ(G)\gamma(G), and the total domination number, γt(G)\gamma_t(G), we consider 13 domination invariants in graphs GG. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure
    corecore