Geodesic
Maximum Independent Sets in Direct Products of Cycles or Trees with Arbitrary Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 675-688

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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.
Keywords: direct product, independent set 
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     title = {Maximum {Independent} {Sets} in {Direct} {Products} of {Cycles} or {Trees} with {Arbitrary} {Graphs}},
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Paj, Tjaša; Špacapan, Simon. Maximum Independent Sets in Direct Products of Cycles or Trees with Arbitrary Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 4, pp. 675-688. http://geodesic.mathdoc.fr/item/DMGT_2015_35_4_a6/