Saturation numbers of the remaining unresolved connected graphs on five vertices

Determine the saturation numbers of the connected five-vertex graphs H5 and K_s^4, where H5 is obtained from C5 by adding one chord and K_s^4 is obtained from K4 by replacing an edge with a path of length two.

Background

Hua and Peng posed the problem of determining saturation numbers for five-vertex graphs whose saturation numbers were not yet known. The paper identifies three connected five-vertex graphs associated with that problem: the kite graph K, H5, and K_s^4. The present paper resolves the kite graph by determining sat(n, K) for all n ≥ 5 and characterizing all extremal graphs, leaving H5 and K_s^4 as the remaining specific unresolved cases from the cited problem.

The graph H5 is defined as the graph obtained from a five-cycle by adding one chord. The graph K_s^4 is defined by replacing one edge of K4 with a path of length two. Determining their saturation numbers would complete the unresolved portion of the cited five-vertex saturation-number problem.