E. Mohr | Searcharxiv

arXiv2018

On the maximum number of maximum independent sets in connected graphs

We characterize the connected graphs of given order $n$ and given independence number $α$ that maximize the number of maximum independent sets. For $3\leq α\leq n/2$, there is a unique such graph that arises from the disjoint union of $α$ cliques of orders $\left\lceil\frac{n}α\right\rceil$ and $\left\lfloor\frac{n}α\right\rfloor$, by selecting a vertex $x$ in a largest clique and adding an edge between $x$ and a vertex in each of the remaining $α-1$ cliques. Our result confirms a conjecture of Derikvand and Oboudi [On the number of maximum independent sets of graphs, Transactions on Combinatorics 3 (2014) 29-36].

math.CO↗

arXiv2018

On the maximum number of minimum dominating sets in forests

Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $γ$ has at most $2^γ$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $γ$ and $2.0598^γ$ minimum dominating sets. We show that every forest with domination number $γ$ has at most $2.4606^γ$ minimum dominating sets, and that every tree with independence number $α$ has at most $2^{α-1}+1$ maximum independent sets.

math.CO↗