# Search results for: Michael A. Henning

Journal of Combinatorial Optimization > 2019 > 38 > 3 > 911-926

*S*of vertices in a graph

*G*is a dominating set if every vertex in $$V(G) {\setminus } S$$ V ( G ) \ S is adjacent to a vertex in

*S*. If the graph

*G*has no isolated vertex, then a semipaired dominating set of

*G*is a dominating set of

*G*with the additional property that the set

*S*can be partitioned into two element subsets such that the vertices in each subset are at most distance two...

Graphs and Combinatorics > 2019 > 35 > 4 > 881-912

*S*of vertices in a graph

*G*is an independent dominating set of

*G*if

*S*is an independent set and every vertex not in

*S*is adjacent to a vertex in

*S*. The independent domination number of

*G*, denoted by

*i*(

*G*), is the minimum cardinality of an independent dominating set. In this paper, we study the following conjecture posed by Goddard and Henning (Discrete Math. 313:839–854, 2013): If $$G\not \cong...

Journal of Combinatorial Optimization > 2019 > 38 > 3 > 721-738

*S*of vertices in a graph

*G*is a dominating set of

*G*if every vertex not in

*S*is adjacent to some vertex in

*S*. The domination number, $$\gamma (G)$$ γ ( G ) , of

*G*is the minimum cardinality of a dominating set of

*G*. Lemańska (Discuss Math Graph Theory 24:165–170, 2004) showed that if

*T*is a tree of order $$n \ge 2$$ n ≥ 2 with $$\ell $$ ℓ leaves, then $$\gamma (T) \ge (n-\ell...

Graphs and Combinatorics > 2018 > 34 > 6 > 1159-1174

Graphs and Combinatorics > 2018 > 34 > 6 > 1371-1384

*G*starts with an initial subset

*S*of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set

*S*is called a forcing set (zero forcing set) of

*G*if, by iteratively applying the forcing process, every...

Journal of Graph Theory > 89 > 2 > 115 - 149

*G*, of order

*n*size

*m*and maximum degree at most

*k*. If

*k*is odd, then ${\alpha}^{\prime}\left(G\right)\ge \left(\frac{k-1}{k({k}^{2}-3)}\right)n\phantom{\rule{0ex}{0ex}}+\phantom{\rule{0ex}{0ex}}\left(\frac{{k}^{2}-k-2}{k({k}^{2}-3)}\right)m\phantom{\rule{0ex}{0ex}}-\phantom{\rule{0ex}{0ex}}\frac{k-1}{k({k}^{2}-3)}$. If

*k*is even, then ${\alpha}^{\prime}\left(G\right)\ge \frac{n}{k(k+1)}\phantom{\rule{0ex}{0ex}}+\phantom{\rule{0ex}{0ex}}\frac{m}{k+1}-\frac{1}{k}$. If

*k*is even, then ${\alpha}^{\prime}\left(G\right)\ge \left(\frac{k+2}{{k}^{2}+k+2}\right)m\phantom{\rule{0ex}{0ex}}-\phantom{\rule{0ex}{0ex}}\left(\frac{k-2}{{k}^{2}+k+2}\right)n\phantom{\rule{0ex}{0ex}}-\frac{k+2}{{k}^{2}+k+2}$. In this article, we actually prove a slight strengthening of the above for which the bounds...

Graphs and Combinatorics > 2018 > 34 > 4 > 819-844

*S*of vertices in a graph

*G*is a dominating set if every vertex in $$V(G) {\setminus } S$$ V(G)\S is adjacent to a vertex in

*S*. If the graph

*G*has no isolated vertex, then a semipaired dominating set of

*G*is a dominating set of

*G*with the additional property that the set

*S*can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The...

Bulletin of the Malaysian Mathematical Sciences Society > 2018 > 41 > 4 > 2141-2149

*G*, and let

*G*|

*v*mean that a vertex

*v*of

*G*is declared to be already totally dominated. A graph

*G*is total domination game critical if $$\gamma _\mathrm{tg}(G|v) < \gamma _\mathrm{tg}(G)$$ γtg(G|v)<γtg(G) holds for every vertex

*v*in

*G*. If $$\gamma _\mathrm{tg}(G) = k$$ γtg(G)=k , then

*G*is further called...

Journal of Combinatorial Optimization > 2018 > 36 > 2 > 416-433

*G*be a graph with vertex set

*V*and no isolated vertices, and let

*S*be a dominating set of

*V*. The set

*S*is a semitotal dominating set of

*G*if every vertex in

*S*is within distance 2 of another vertex of

*S*. And,

*S*is a semipaired dominating set of

*G*if

*S*can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semitotal domination number $$\gamma...

Journal of Graph Theory > 88 > 1 > 174 - 191

Journal of Graph Theory > 88 > 1 > 131 - 145

*d*of nonnegative integers, let $G\left(d\right)$ and $F\left(d\right)$ be the sets of all graphs and forests with degree sequence

*d*, respectively. Let ${\gamma}_{min}\left(d\right)=min\{\gamma \left(G\right):G\in G\left(d\right)\}$, ${\alpha}_{max}\left(d\right)=max\{\alpha \left(G\right):G\in G\left(d\right)\}$, ${\gamma}_{min}^{F}\left(d\right)=min\{\gamma \left(F\right):F\in F\left(d\right)\}$, and ${\alpha}_{max}^{F}\left(d\right)=max\{\alpha \left(F\right):F\in F\left(d\right)\}$ where $\gamma \left(G\right)$ is the domination number and $\alpha \left(G\right)$ is the independence number of a graph

*G*. Adapting results of Havel and Hakimi, Rao showed in 1979 that ${\alpha}_{max}$...

Discrete Applied Mathematics > 2018 > 236 > C > 246-255

Discrete Applied Mathematics > 2018 > 236 > C > 235-245

Discrete Applied Mathematics > 2018 > 236 > C > 66-72

Discussiones Mathematicae Graph Theory > 2017 > 38 > 1 > 203-215

Discussiones Mathematicae Graph Theory > 2017 > 38 > 1 > 275-285

Discrete Mathematics > 2018 > 341 > 1 > 155-164

Graphs and Combinatorics > 2018 > 34 > 1 > 261-276

*G*is a set

*S*of vertices of

*G*such that every vertex not in

*S*has a neighbor in

*S*. Further, if every vertex of

*G*has a neighbor in

*S*, then

*S*is a total dominating set of

*G*. The domination number, $$\gamma (G)$$ γ(G) , and total domination number, $$\gamma _{t}(G)$$ γt(G) , are the minimum cardinalities of a dominating set and total dominating set, respectively, in

*G*. The...

Discrete Mathematics > 2017 > 340 > 10 > 2349-2354

Information Processing Letters > 2017 > 126 > C > 12-17