P3P3-equicoverable graphs—Research on HH-equicoverable graphs

Let HH be a fixed graph. An HH-covering   of GG is a set L={H1,H2,…,Hk}L={H1,H2,…,Hk} of subgraphs of GG, where each subgraph HiHi is isomorphic to HH and every edge of GG appears in at least one member of LL. If there exists an HH-covering of GG, GG is called HH-coverable  . An HH-covering of GG with kk copies H1,H2,…,HkH1,H2,…,Hk of HH is called minimal   if, for any HjHj, ⋃i=1kHi-Hj is not an HH-covering of GG. An HH-covering of GG with kk copies H1,H2,…,HkH1,H2,…,Hk of HH is called minimum   if there exists no HH-covering with less than kk copies of H  . A graph GG is called HH-equicoverable   if every minimal HH-covering in GG is also a minimum HH-covering in GG. In this paper, we investigate the characterization of P3P3-equicoverable graphs.