A superclass of Edge-Path-Tree graphs with few cliques

Edge-Path-Tree (EPT) graphs are intersection graphs of EPT matrices that is matrices whose columns are incidence vectors of edge-sets of paths in a given tree. EPT graphs have polynomially many cliques [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinational Theory Series B 38 (1985) 8-22; C.L. Monma, V.K. Wey, Intersection graphs of paths in a tree, Journal of Combinational Theory Series B 41 (1986) 141-181]. Therefore, the problem of finding a clique of maximum weight in these graphs is solvable in strongly polynomial time. We extend this result to a proper superclass of EPT graphs.