586 research outputs found

    Locating-dominating sets in twin-free graphs

    Full text link
    A locating-dominating set of a graph GG is a dominating set DD of GG with the additional property that every two distinct vertices outside DD have distinct neighbors in DD; that is, for distinct vertices uu and vv outside DD, N(u)∩D≠N(v)∩DN(u) \cap D \ne N(v) \cap D where N(u)N(u) denotes the open neighborhood of uu. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of GG, denoted γL(G)\gamma_L(G), is the minimum cardinality of a locating-dominating set in GG. It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] that if GG is a twin-free graph of order nn without isolated vertices, then γL(G)≤n2\gamma_L(G)\le \frac{n}{2}. We prove the general bound γL(G)≤2n3\gamma_L(G)\le \frac{2n}{3}, slightly improving over the ⌊2n3⌋+1\lfloor\frac{2n}{3}\rfloor+1 bound of Garijo et al. We then provide constructions of graphs reaching the n2\frac{n}{2} bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees GG that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.Comment: 11 pages; 4 figure

    On perfect and quasiperfect dominations in graphs

    Get PDF
    A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).Postprint (published version

    On the extremal properties of the average eccentricity

    Get PDF
    The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G)ecc (G) of a graph GG is the mean value of eccentricities of all vertices of GG. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease ecc(G)ecc (G). Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
    • …
    corecore