On the restricted size Ramsey number involving a path P_3

For any pair of graphs G and H, both the size Ramsey number ̂r(G,H) and the restricted size Ramsey number r*(G,H) are bounded above by the size of the complete graph with order equals to the Ramsey number r(G,H), and bounded below by e(G) + e(H) − 1. Moreover, trivially, ̂r(G,H) ≤ r*(G,H). When introducing the size Ramsey number for graph, Erdős et al. (1978) asked two questions; (1) Do there exist graphs G and H such that ˆr(G,H) attains the upper bound? and (2) Do there exist graphs G and H such that ̂r(G,H) is significantly less than the upper bound? In this paper we consider the restricted size Ramsey number r*(G,H). We answer both questions above for r*(G,H) when G = P3 and H is a connected graph.