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2-clique circular arc adjacent to vertex alarm algorithm assigning red asteroidal asteroidal triple AVL tree bipartite graph bipartite probe interval blue vertex characterization chromatic number circular arc graphs Color all vertices complement conjecture connected graph copwin graph CW complex defined Definition delete denote diametric path F-coloring Figure freepath Graffiti.pc graph G Graph Theory graphs of order hamilton paintable Hasse diagram homogeneous paintable HQ(n induce a forest induced subgraph integer intersection graphs interval bigraphs interval point bigraphs inversion tables Lemma Let G Math maximum minimum number minimum spanning tree O-arrays optimal file transfer outdegree outdegree sequence permutation planar graph polynomial poset probe interval graphs Proof Proposition red vertex red-blue coloring remaining vertices result root node SEAM number stars Steinbach Steinbach numbers structure subgraph subtree Theorem thickened graph unit interval unit interval bigraphs vertex set vertices of G Vk-i Vk+2 Vk+i XQ(n