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3-space achiral acyclic algorithm axial bipartite graph Cayley CF-graph chiral chordal graphs chromatic number circuits clique cographs collapsible coloring complete graph complete subgraph Computer conjecture connected graph contains Corollary covering number cycle cycle-minimal graph defined degree linear degree representative denote dicycles digraph disjoint edge of G edge subchromatic number equations equivalent eulerian Eulerian Graphs example finite graph algebra graph G graph obtained graph of order Graph Theory hamiltonian Harary induced subgraph integer irregular network isomorphic K-tree Kirchhoff knot lattice Lemma Let G Math mathematical maximum number multigraph network reliability nonplanar nontrivial number of edges number of vertices Outerplanar Graphs paper parameters partial fc-trees partition path permutations planar graphs polynomial problem proof Proposition rotational symmetries sequence series-parallel graphs shown in Figure spanning trees spatial embedding subcoloring subgraph of G subset Theorem theta curve topologically ultimately economical vertex vertices of degree vertices of G