651 research outputs found

    Domination in Functigraphs

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    Let G1G_1 and G2G_2 be disjoint copies of a graph GG, and let f:V(G1)→V(G2)f: V(G_1) \rightarrow V(G_2) be a function. Then a \emph{functigraph} C(G,f)=(V,E)C(G, f)=(V, E) has the vertex set V=V(G1)∪V(G2)V=V(G_1) \cup V(G_2) and the edge set E=E(G1)∪E(G2)∪{uv∣u∈V(G1),v∈V(G2),v=f(u)}E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let γ(G)\gamma(G) denote the domination number of GG. It is readily seen that γ(G)≤γ(C(G,f))≤2γ(G)\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G). We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.Comment: 18 pages, 8 figure

    Upper paired domination versus upper domination

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    A paired dominating set PP is a dominating set with the additional property that PP has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph GG is called the upper domination number of GG, denoted by Γ(G)\Gamma(G), the maximum cardinality of a minimal paired dominating set in GG is called the upper paired domination number of GG, denoted by Γpr(G)\Gamma_{pr}(G). By Henning and Pradhan (2019), we know that Γpr(G)≤2Γ(G)\Gamma_{pr}(G)\leq 2\Gamma(G) for any graph GG without isolated vertices. We focus on the graphs satisfying the equality Γpr(G)=2Γ(G)\Gamma_{pr}(G)= 2\Gamma(G). We give characterizations for two special graph classes: bipartite and unicyclic graphs with Γpr(G)=2Γ(G)\Gamma_{pr}(G)= 2\Gamma(G) by using the results of Ulatowski (2015). Besides, we study the graphs with Γpr(G)=2Γ(G)\Gamma_{pr}(G)= 2\Gamma(G) and a restricted girth. In this context, we provide two characterizations: one for graphs with Γpr(G)=2Γ(G)\Gamma_{pr}(G)= 2\Gamma(G) and girth at least 6 and the other for C3C_3-free cactus graphs with Γpr(G)=2Γ(G)\Gamma_{pr}(G)= 2\Gamma(G). We also pose the characterization of the general case of C3C_3-free graphs with Γpr(G)=2Γ(G)\Gamma_{pr}(G)= 2\Gamma(G) as an open question

    Disjoint Dominating Sets with a Perfect Matching

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    In this paper, we consider dominating sets DD and D′D' such that DD and D′D' are disjoint and there exists a perfect matching between them. Let DDm(G)DD_{\textrm{m}}(G) denote the cardinality of smallest such sets D,D′D, D' in GG (provided they exist, otherwise DDm(G)=∞DD_{\textrm{m}}(G) = \infty). This concept was introduced in [Klostermeyer et al., Theory and Application of Graphs, 2017] in the context of studying a certain graph protection problem. We characterize the trees TT for which DDm(T)DD_{\textrm{m}}(T) equals a certain graph protection parameter and for which DDm(T)=α(T)DD_{\textrm{m}}(T) = \alpha(T), where α(G)\alpha(G) is the independence number of GG. We also further study this parameter in graph products, e.g., by giving bounds for grid graphs, and in graphs of small independence number

    On the existence of total dominating subgraphs with a prescribed additive hereditary property

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    AbstractRecently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set
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