Let \(P_t\) and \(C_t\) denote the path and the cycle on t vertices, respectively. A diamond (resp. gem) consists of a \(P_3\) (resp. \(P_4\) ) and a new vertex adjacent to all vertices of the \(P_3\) (resp. \(P_4\) ), a kite consists of a \(P_4\) and a new vertex adjacent to three consecutive vertices of the \(P_4\) , and a paraglider consists of a \(C_4\) and a new vertex adjacent to three vertices of the \(C_4\) . A class \(\mathcal{G}\) of graphs is said to be \(\chi \) -polydet (Schiermeyer and Randerath in Graphs Combin 35:1–31, 2019) if \(\mathcal{G}\) has a polynomial binding function f and there exists a polynomial time algorithm to determine a coloring of \(G\in \mathcal{G}\) with at most \(f(\omega (G))\) colors. In 2021, Choudum et al. (Disc. Math. 344:112244, 2021) determined the structures of \((P_7,C_7,C_4\) , diamond)-free and \((P_7,C_7,C_4\) , gem)-frees, and gave correspondingly tight upper bounds to the chromatic numbers of these graphs. In this paper, we study the structure of \((P_7, C_5\) , kite, paraglider)-free graphs, which is a superfamily of \((P_7, C_5\) , diamond)-free graphs. We show that there is a unique connected imperfect \((P_7, C_5\) , kite, paraglider)-free graph with \(\delta (G)\ge \omega (G)+1\) , which has no clique cutsets, no universal cliques, and no pair of vertices of which one’s neighborhoods contains the other’s. As a consequence, we show that \((P_7, C_5\) , kite, paraglider)-free graphs are \(\chi \) -polydet with a binding function \(\omega (G)+1\) .
