Forbidden Subgraphs for Chorded Pancyclicity

We call a graph $G$ pancyclic if it contains at least one cycle of every possible length $m$, for $3\le m\le \mathrm{\mid }V(G)\mathrm{\mid }$. In this paper, we define a new property called chorded pancyclicity. We explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains at least one chorded cycle of every possible length $4,5,\dots ,\mathrm{\mid }V(G)\mathrm{\mid }$. In particular, certain paths and triangles with pendant paths are forbidden.