Assertion: graph's g | /r/HasProperty | triangle-free
Facets:
Source sentences:
text: In one direction, suppose one is given as input a graph G, and must test whether G is triangle-free.
text: H is by construction a partial cube, but it is a median graph only when G is triangle-free: if a, b, and c form a triangle in G, then {a,b}, {a,c}, and {b,c} have no median in H, for such a median would have to correspond to the set {a,b,c}, but sets of three or more vertices of G do not form vertices in H.
text: Therefore, G is triangle-free
text: In the case that G is triangle-free, H is its simplex graph.
text: An algorithm to test efficiently whether H is a median graph could by this construction also be used to test whether G is triangle-free.