Maximum Independent Sets in Direct Products of Cycles or Trees with Arbitrary Graphs - Paj Tjaša; Špacapan Simon (Discussiones Mathematicae Graph Theory, 2015)
The direct product of graphs G = (V (G),E(G)) and H = (V (H),E(H)) is the graph, denoted as G×H, with vertex set V (G×H) = V (G)×V (H), where vertices (x1, y1) and (x2, y2) are adjacent in G × H if x1x2 ∈ E(G) and y1y2 ∈ E(H). Let n be odd and m even. We prove that every maximum independent set in Pn×G, respectively Cm×G, is of the form (A×C)∪(B× D), where C and D are nonadjacent in G, and A∪B is the bipartition of Pn respectively Cm. We also give a characterization of maximum independent subsets of Pn × G for every even n and discuss the structure of maximum independent sets in T × G where T is a tree.