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Bounds on the semipaired domination number of graphs with minimum degree at least two
Journal of Combinatorial Optimization ( IF 0.9 ) Pub Date : 2021-01-19 , DOI: 10.1007/s10878-020-00687-w
Teresa W. Haynes , Michael A. Henning

Let G be a graph with vertex set V and no isolated vertices. A subset \(S \subseteq V\) is a semipaired dominating set of G if every vertex in \(V {\setminus } S\) is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number \(\gamma _\mathrm{pr2}(G)\) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph of order n with minimum degree at least 2, then \(\gamma _\mathrm{pr2}(G) \le \frac{1}{2}(n+1)\). Further, we show that if \(n \not \equiv 3 \, (\mathrm{mod}\, 4)\), then \(\gamma _\mathrm{pr2}(G) \le \frac{1}{2}n\), and we show that for every value of \(n \equiv 3 \, (\mathrm{mod}\, 4)\), there exists a connected graph G of order n with minimum degree at least 2 satisfying \(\gamma _\mathrm{pr2}(G) = \frac{1}{2}(n+1)\). As a consequence of this result, we prove that every graph G of order n with minimum degree at least 3 satisfies \(\gamma _\mathrm{pr2}(G) \le \frac{1}{2}n\).



中文翻译:

最小度至少为2的图的半对控制数上的界

G为顶点集为V且没有孤立顶点的图。一个子集\(S \ subseteq V \)是一个semipaired支配集ģ如果在每个顶点\(V {\ setminus}Š\)相邻于一个顶点小号小号可以划分为两个元件的子集,使得所述每个子集中的顶点最多相距两个距离。所述semipaired控制数\(\伽马_ \ mathrm {PR2}(G)\)是一个semipaired支配集的最小基数ģ。我们证明,如果G是阶n且最小度至少为2的连通图 ,则\(\ gamma _ \ mathrm {pr2}(G)\ le \ frac {1} {2}(n + 1)\)。此外,我们证明如果\(n \ not \ equiv 3 \,(\ mathrm {mod} \ ,, 4)\),则\(\ gamma _ \ mathrm {pr2}(G)\ le \ frac {1} {2} n \),我们证明对于\(n \ equiv 3 \,(\ mathrm {mod} \,4)\)的每个值,存在一个阶n的连通图G,其最小度至少为2满足\(\ gamma _ \ mathrm {pr2}(G)= \ frac {1} {2}(n + 1)\)。作为该结果的结果,我们证明最小阶n的每个n阶 图G至少满足\(\ gamma _ \ mathrm {pr2}(G)\ le \ frac {1} {2} n \)

更新日期:2021-01-20
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