61 research outputs found

    Injective coloring of product graphs

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    The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph coloring and domination theory. We make use of these three points of view in this paper so as to investigate the injective coloring of some well-known graph products. We bound the injective chromatic number of direct and lexicographic product graphs from below and above. In particular, we completely determine this parameter for the direct product of two cycles. We also give a closed formula for the corona product of two graphs

    Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond

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    A dominating set SS of graph GG is called an rr-grouped dominating set if SS can be partitioned into S1,S2,,SkS_1,S_2,\ldots,S_k such that the size of each unit SiS_i is rr and the subgraph of GG induced by SiS_i is connected. The concept of rr-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r=1r=1), paired dominating sets (r=2r=2), and connected dominating sets (rr is arbitrary and k=1k=1). In this paper, we investigate the computational complexity of rr-Grouped Dominating Set, which is the problem of deciding whether a given graph has an rr-grouped dominating set with at most kk units. For general rr, the problem is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which rr is a constant or a parameter, but we see that the problem for every fixed r>0r>0 is still hard to solve. From the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that it is fixed-parameter tractable for rr and treewidth, because the condition of rr-grouped domination for a constant rr can be represented as monadic second-order logic (mso2). This is good news, but the running time is not practical. We then design an O(min{(2τ(r+1))τ,(2τ)2τ})O^*(\min\{(2\tau(r+1))^{\tau},(2\tau)^{2\tau}\})-time algorithm for general r2r\ge 2, where τ\tau is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r{2,3}r \in \{2,3\}, we can speed up the algorithm, whose running time becomes O((r+1)τ)O^*((r+1)^\tau). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of rr-Grouped Dominating Set.Comment: 23 pages, 6 figure

    2022: The President\u27s Dinner For Faculty and Staff

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    Tammy and I welcome you to this 72nd President’s Dinner! That’s a long tradition. Just think of how many professional forefathers and mothers held the position you now occupy. Think of their steadfast dedication to our great cause, their grit during challenge, their timely talent consecrated to God through this place and for these people. Thank you for carrying the torch forward, for our time. Tonight we enjoy the theme, “A Fruitful Community.” The tense and sour American culture makes this Olivet community, by contrast, quite life-giving! Yet imagine an even deeper joy as we become an especially fruitful community. Our special thanks to Prof. Ashley Sarver for skillfully hosting the program, our Resident Assistants for graciously serving the meal, and to special guests like retired faculty who honor us by their presence. Enjoy the evening! ___ Gregg and Tammy Chenowet

    Advances in Discrete Applied Mathematics and Graph Theory

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    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

    Distances and Domination in Graphs

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    This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

    Acta Universitatis Sapientiae - Informatica 2020

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    Domination parameters with number 2: Interrelations and algorithmic consequences

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    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
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