in this example, the edges going upward, to the left, and to the right from the central degree-four vertex do not have any cliques in common. therefore, any partition of the graph's edges into cliques would have to have at least one clique for each of these three edges, and these three cliques would all intersect in that central vertex, violating the requirement that each vertex appear in exactly two cliques. thus, the graph shown is not a line graph.