226 research outputs found

    On the approximability of the maximum induced matching problem

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    In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3

    Cubic graphs with large circumference deficit

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    The circumference c(G)c(G) of a graph GG is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 44-, 55- and 66-edge-connected cubic graphs with circumference ratio c(G)/V(G)c(G)/|V(G)| bounded from above by 0.8760.876, 0.9600.960 and 0.9900.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 44-edge-connected cubic graph is at least 0.750.75. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544]

    Edge Roman domination on graphs

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    An edge Roman dominating function of a graph GG is a function f ⁣:E(G){0,1,2}f\colon E(G) \rightarrow \{0,1,2\} satisfying the condition that every edge ee with f(e)=0f(e)=0 is adjacent to some edge ee' with f(e)=2f(e')=2. The edge Roman domination number of GG, denoted by γR(G)\gamma'_R(G), is the minimum weight w(f)=eE(G)f(e)w(f) = \sum_{e\in E(G)} f(e) of an edge Roman dominating function ff of GG. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if GG is a graph of maximum degree Δ\Delta on nn vertices, then γR(G)ΔΔ+1n\gamma_R'(G) \le \lceil \frac{\Delta}{\Delta+1} n \rceil. While the counterexamples having the edge Roman domination numbers 2Δ22Δ1n\frac{2\Delta-2}{2\Delta-1} n, we prove that 2Δ22Δ1n+22Δ1\frac{2\Delta-2}{2\Delta-1} n + \frac{2}{2\Delta-1} is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of kk-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on nn vertices is at most 67n\frac{6}{7}n, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain K2,3K_{2,3} as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs
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