Question

A dominating set of vertices in a simple graph is a set of vertices such that every other vertex is adjacent to at least one vertex of this set. A dominating set with the least number of vertices is called a minimum dominating set. Find a minimum dominating set for the given graph.

Short answer

The minimum dominating set for the graph is \(\left\{ {c,d} \right\}\).

Step-by-step solution

Concept/Significance of a minimum dominating set

A minimum dominating set is one that isn't a legitimate subset of some other dominating set in a graph. The domatic number of a graph can be calculated with minimal dominant sets.

Determination of the minimum dominating set in the graph

As each of the other vertices, a, b, e, and f, is next to at least one of the cords,c and d,\(\left\{ {c,d} \right\}\)is the smallest dominating set, implying that there is no other smaller dominating set.

Thus, the minimum dominating set for the given graph is \(\left\{ {c,d} \right\}\).