Unilateral Orientation of Mixed Graphs Tamara Mchedlidze, Antonios - PowerPoint PPT Presentation

Unilateral Orientation of Mixed Graphs Tamara Mchedlidze, Antonios Symvonis Dept. of Mathematics, National Technical University of Athens, Athens, Greece. { mchet,symvonis } @math.ntua.gr January 25, 2010 T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 1 / 11

Definitions Strong Digraph Unilateral Digraph 2 2 3 3 1 1 4 4 Strong digraph Unilateral digraph T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 2 / 11

Definitions Mixed Graph Strong Orientation Unilateral Orientation v v v 1 1 1 v v v 2 2 2 v v v 4 4 4 v v v 3 3 3 Strong orientation of G Unilateral orientation of G Mixed graph G T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 3 / 11

Definitions Problem: Given a mixed graph G , determine if G has a strong or a unilateral orientation. v v v 1 1 1 v v v 2 2 2 v v v 4 4 4 v v v 3 3 3 Strong orientation of G Unilateral orientation of G Mixed graph G T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 3 / 11

Some known results Theorem (Robbins, 1939) A connected graph G has a strongly connected orientation if and only if G has no bridge. Theorem (Boesch and Tindell, 1980) A mixed multigraph M admits a strong orientation if and only if M is strong and the underlying multigraph of M is bridgeless. 2 8 11 7 3 5 1 9 6 10 4 T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 4 / 11

Some known results Theorem (Chartrand, Harary, Schultz, Wall, 1994) A connected graph G has a unilateral orientation if and only if all of the bridges of G lie on a common path. Question: What about unilateral orientations of mixed graphs? 2 8 11 7 3 5 1 9 6 10 4 T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 4 / 11

Some known results Theorem (Chartrand, Harary, Schultz, Wall, 1994) A connected graph G has a unilateral orientation if and only if all of the bridges of G lie on a common path. Question: What about unilateral orientations of mixed graphs? Open till now! 2 8 11 7 3 5 1 9 6 10 4 T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 4 / 11

Main Results Characterization of Unilaterally Orientable Mixed Graphs Linear Time Algorithm Testing if a Given Mixed Graph has a Unilateral Orientation T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 5 / 11

Characterization of unilaterally orienable mixed graphs M 2 13 12 14 1 3 31 16 30 4 15 29 32 Strong component digraph of M 33 18 17 m 7 m m 2 3 1 5 19 27 26 20 6 (b) 25 28 8 11 24 23 9 22 21 10 M M M 2 3 1 (a) Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ the strong component digraph of M, SC ( M ) , has a hamiltonian path. T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 6 / 11

Characterization of unilaterally orienable mixed graphs Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC ( M ) , has a hamiltonian path. V v a SC(M) M b u U Vertices a and b are not connected Stong component digraph without hamiltonian path by a directed path Sketch of proof: 1 Assume that there is no hamiltonian path in SC ( M ) 2 SC ( M ) is acyclic digraph T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 7 / 11

Characterization of unilaterally orienable mixed graphs Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC ( M ) , has a hamiltonian path. V v a SC(M) M b u U Vertices a and b are not connected Stong component digraph without hamiltonian path by a directed path Sketch of proof: 3 There are two vertices in SC ( M ), u and v , that are not connected by a directed path in either direction T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 7 / 11

Characterization of unilaterally orienable mixed graphs Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC ( M ) , has a hamiltonian path. V v a SC(M) M b u U Vertices a and b are not connected Stong component digraph without hamiltonian path by a directed path Sketch of proof: 4 Corresponding strong components of the biorientation of M , U and V contain vertices a and b that are not connected by a directed path. T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 7 / 11

Characterization of unilaterally orienable mixed graphs Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC ( M ) , has a hamiltonian path. V v a SC(M) M b u U Vertices a and b are not connected Stong component digraph without hamiltonian path by a directed path Sketch of proof: 5 M do not admit a unilateral orientation. T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 7 / 11

Characterization of unilaterally orienable mixed graphs M 2 13 m 2,1 12 14 1 m 3 1,1 Bridge graphs m 3,1 31 16 30 15 4 32 29 m m 33 2,1 18 17 1,1 m 7 1,2 m 3,1 5 m m 2,2 1,2 27 26 19 20 6 m 2,2 m 3,2 m m m 1,3 2,3 1,3 28 25 m 3,2 8 11 24 Bridgeless−component mixed graph: BC(M) m 2,3 23 9 22 21 10 M M M 1 2 3 (a) (b) Lemma (Second necessary condition) If a mixed graph M admits a unilateral orientation ⇒ the bridge graph, B ( M ) , of each of its strong component is a path. T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 8 / 11

Characterization of unilaterally orienable mixed graphs M 2 13 m 2,1 12 14 1 m 3 1,1 Bridge graphs m 3,1 31 16 30 15 4 32 29 m m 33 2,1 18 17 1,1 m 7 1,2 m 3,1 5 m m 2,2 1,2 27 26 19 20 6 m 2,2 m 3,2 m m m 1,3 2,3 1,3 28 25 m 3,2 8 11 24 Bridgeless−component mixed graph: BC(M) m 2,3 23 9 22 21 10 M M M 1 2 3 (a) (b) Theorem (Main result) A mixed graph M admits a unilateral orientation ⇔ the bridgeless-component mixed graph, BC ( M ) , admits a hamiltonian orientation. T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 8 / 11

Characterization of unilaterally orienable mixed graphs M 2 13 m 2,1 12 14 1 m 3 1,1 Bridge graphs m 3,1 31 16 30 15 4 32 29 m m 33 2,1 18 17 1,1 m 7 1,2 m 3,1 5 m m 2,2 1,2 27 26 19 20 6 m 2,2 m 3,2 m m m 1,3 2,3 1,3 28 25 m 3,2 8 11 24 Bridgeless−component mixed graph: BC(M) m 2,3 23 9 22 21 10 M M M 1 2 3 (a) (b) Sketch of proof: ( M unilaterally orientable ⇒ BC ( M ) has a hamiltonian orientation) Unilateral orientation of M contains a spanning walk C C induces a hamiltonian path on BC ( M ) T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 8 / 11

Characterization of unilaterally orienable mixed graphs M 2 13 m 2,1 12 14 1 m 3 1,1 Bridge graphs m 3,1 31 16 30 15 4 32 29 m m 33 2,1 18 17 1,1 m 7 1,2 m 3,1 5 m m 2,2 1,2 27 26 19 20 6 m 2,2 m 3,2 m m m 1,3 2,3 1,3 28 25 m 3,2 8 11 24 Bridgeless−component mixed graph: BC(M) m 2,3 23 9 22 21 10 M M M 1 2 3 (a) (b) Sketch of proof: ( M unilaterally orientable ⇐ BC ( M ) has a hamiltonian orientation) Each vertex of BC ( M ) admits a strong orientation That together with hamiltonian orientation of BC ( M ) gives a unilateral orientation T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 8 / 11

Characterization of unilaterally orienable mixed graphs M 2 13 m 2,1 12 14 1 m 3 1,1 Bridge graphs m 3,1 31 16 30 15 4 32 29 m m 33 2,1 18 17 1,1 m 7 1,2 m 3,1 5 m m 2,2 1,2 27 26 19 20 6 m 2,2 m 3,2 m m m 1,3 2,3 1,3 28 25 m 3,2 8 11 24 Bridgeless−component mixed graph: BC(M) m 2,3 23 9 22 21 10 M M M 1 2 3 (a) (b) Sketch of proof: ( M unilaterally orientable ⇐ BC ( M ) has a hamiltonian orientation) Each vertex of BC ( M ) admits a strong orientation That together with hamiltonian orientation of BC ( M ) gives a unilateral orientation T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 9 / 11

Summerizing Theorem Given a mixed graph M = ( V , A , E ) , we can decide whether M admits a unilateral orientation in O ( V + A + E ) time. Moreover, if M is unilaterally orientable, a unilateral orientation can be computed in O ( V + A + E ) time. T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 10 / 11

Thank You! T. Mchedlidze, A. Symvonis Dept. of Mathematics, National Technical University of Athens. 11 / 11