Abstract

In this note, we give a simpler proof of a theorem of Desormeaux et al., which states that for any connected graph G containing a cycle, γc(G)≥g(G)−2, where γc(G) and g(G) are the connected domination number and the girth of G respectively. This enables us to confirm a conjecture of Walsh: for any connected nontree graph G, h(G)≥g(G)−3, where h(G) is the hub number of G. We also determine the connected domination number and the connected hub number of Mycielski graphs.

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