1. Graphs with equal Grundy domination and independence numberGábor Bacsó, Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall, 2023, original scientific article Abstract: The Grundy domination number, ${\gamma_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $jKeywords: Grundy domination, independence number, upper domination number, bipartite graphs
Published in DiRROS: 09.04.2024; Views: 80; Downloads: 44
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2. Resonance graphs and a binary coding of perfect matchings of outerplane bipartite graphsSimon Brezovnik, Niko Tratnik, Petra Žigert Pleteršek, 2023, original scientific article Abstract: The aim of this paper is to investigate resonance graphs of $2$-connected outerplane bipartite graphs, which include various families of molecular graphs. Firstly, we present an algorithm for a binary coding of perfect matchings of these graphs. Further, $2$-connected outerplane bipartite graphs with isomorphic resonance graphs are considered. In particular, it is shown that if two $2$-connected outerplane bipartite graphs are evenly homeomorphic, then its resonance graphs are isomorphic. Moreover, we prove that for any $2$-connected outerplane bipartite graph $G$ there exists a catacondensed even ring systems $H$ such that the resonance graphs of $G$ and $H$ are isomorphic. We conclude with the characterization of $2$-connected outerplane bipartite graphs whose resonance graphs are daisy cubes.
Keywords: graph theory, resonance graphs, bipartite graphs
Published in DiRROS: 18.03.2024; Views: 81; Downloads: 32
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3. Computational complexity aspects of super dominationCsilla Bujtás, Nima Ghanbari, Sandi Klavžar, 2023, original scientific article Abstract: Let ▫$G$▫ be a graph. A dominating set ▫$D\subseteq V(G)$▫ is a super dominating set if for every vertex ▫$x\in V(G) \setminus D$▫ there exists ▫$y\in D$▫ such that ▫$N_G(y)\cap (V(G)\setminus D)) = \{x\}$▫. The cardinality of a smallest super dominating set of ▫$G$▫ is the super domination number of ▫$G$▫. An exact formula for the super domination number of a tree ▫$T$▫ is obtained, and it is demonstrated that a smallest super dominating set of ▫$T$▫ can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph ▫$G$▫ is at most a given integer if ▫$G$▫ is a bipartite graph of girth at least ▫$8$▫. The super domination number is determined for all ▫$k$▫-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determine.
Keywords: super domination number, trees, bipartite graphs, k-subdivision of a graph, computational complexity, matching, II-matching number
Published in DiRROS: 14.03.2024; Views: 96; Downloads: 56
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4. Outerplane bipartite graphs with isomorphic resonance graphsSimon Brezovnik, Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek, 2024, original scientific article Abstract: We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Three additional characterizations are expressed in terms of resonance digraphs, via local structures of inner duals, as well as using distributive lattices on the set of order ideals of posets defined on inner faces of 2-connected outerplane bipartite graphs.
Keywords: distributive lattice, inner dual, isomorphic resonance graphs, order ideal, 2-connected outerplane bipartite graph
Published in DiRROS: 13.03.2024; Views: 96; Downloads: 55
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