2,581 research outputs found

    On The b-Chromatic Number of Regular Graphs Without 4-Cycle

    Full text link
    The b-chromatic number of a graph GG, denoted by ϕ(G)\phi(G), is the largest integer kk that GG admits a proper kk-coloring such that each color class has a vertex that is adjacent to at least one vertex in each of the other color classes. We prove that for each dd-regular graph GG which contains no 4-cycle, ϕ(G)≥⌊d+32⌋\phi(G)\geq\lfloor\frac{d+3}{2}\rfloor and if GG has a triangle, then ϕ(G)≥⌊d+42⌋\phi(G)\geq\lfloor\frac{d+4}{2}\rfloor. Also, if GG is a dd-regular graph which contains no 4-cycle and diam(G)≥6diam(G)\geq6, then ϕ(G)=d+1\phi(G)=d+1. Finally, we show that for any dd-regular graph GG which does not contain 4-cycle and κ(G)≤d+12\kappa(G)\leq\frac{d+1}{2}, ϕ(G)=d+1\phi(G)=d+1

    The effect of girth on the kernelization complexity of Connected Dominating Set

    Get PDF
    In the Connected Dominating Set problem we are given as input a graph GG and a positive integer kk, and are asked if there is a set SS of at most kk vertices of GG such that SS is a dominating set of GG and the subgraph induced by SS is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set. Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer kk (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function g(k)g(k). The new instance is called a g(k)g(k) kernel for the problem. If g(k)g(k) is a polynomial in kk then we say that the problem admits polynomial kernels. The girth of a graph GG is the length of a shortest cycle in GG. It turns out that Connected Dominating Set is ``hard\u27\u27 on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth 33 or 44 (since the problem is W[2]-hard); (b) admits a g(k)g(k) kernel, where g(k)g(k) is kO(k)k^{O(k)}, on graphs of girth 55 or 66 but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic (O(k3)O(k^3)) kernel on graphs of girth at least 77. While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs

    On the complexity of computing the kk-restricted edge-connectivity of a graph

    Full text link
    The \emph{kk-restricted edge-connectivity} of a graph GG, denoted by λk(G)\lambda_k(G), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least kk vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing λk(G)\lambda_k(G). Very recently, in the parameterized complexity community the notion of \emph{good edge separation} of a graph has been defined, which happens to be essentially the same as the kk-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.Comment: 16 pages, 4 figure

    On the Connectivity of Token Graphs of Trees

    Full text link
    Let kk and nn be integers such that 1≤k≤n−11\leq k \leq n-1, and let GG be a simple graph of order nn. The kk-token graph Fk(G)F_k(G) of GG is the graph whose vertices are the kk-subsets of V(G)V(G), where two vertices are adjacent in Fk(G)F_k(G) whenever their symmetric difference is an edge of GG. In this paper we show that if GG is a tree, then the connectivity of Fk(G)F_k(G) is equal to the minimum degree of Fk(G)F_k(G)
    • …
    corecore