Finding compatible circuits in eulerian digraphs James Carraher - PowerPoint PPT Presentation

Finding compatible circuits in eulerian digraphs James Carraher University of Nebraska – Lincoln s-jcarrah1@math.unl.edu Joint Work with Stephen Hartke June 13, 2013 James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Introduction Eulerian digraphs Def. An eulerian digraph G is a digraph that contains a closed walk that visits each edge exactly once. Thm. A digraph G is eulerian if and only if de g − (  ) = deg + (  ) for all vertices  and G is strongly (weakly) connected. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Introduction Eulerian digraphs Def. An eulerian digraph G is a digraph that contains a closed walk that visits each edge exactly once. Thm. A digraph G is eulerian if and only if deg − (  ) = deg + (  ) for all vertices  and G is strongly (weakly) connected. 2 3 1 9 6 7 8 10 11 4 5 James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Compatible Circuits Compatible circuits Def. A colored eulerian digraph G is an eulerian digraph with a fixed edge coloring (not necessarily proper). A compatible circuit is an eulerian circuit of G such that no two consecutive edges in the tour have the same color (i.e. no monochromatic transitions). Good Bad James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Compatible Circuits Compatible circuits Def. A colored eulerian digraph G is an eulerian digraph with a fixed edge coloring (not necessarily proper). A compatible circuit is an eulerian circuit of G such that no two consecutive edges in the tour have the same color (i.e. no monochromatic transitions). 1 9 Good Good 2 4 8 3 Bad Bad 7 5 6 James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Compatible Circuits Applications Eulerian digraphs are applied to routing problems such as garbage collecting, mail carriers, etc. Restrictions on routes such as no U-turns can be modeled by compatible circuits. Other applications: universal cycles of permutations, etc. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Compatible Circuits Examples Big Question: When does an colored eulerian digraph have a compatible cir cuit? Not all graphs have compatible circuits. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Simple necessary condition Let γ (  ) be the size of the largest color class incident to  . James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Simple necessary condition Let γ (  ) be the size of the largest color class incident to  . Prop. If there exists a vertex  where γ (  ) > d eg + (  ) , then G does not have a compatible circuit. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Simple necessary condition Let γ (  ) be the size of the largest color class incident to  . Prop. If there exists a vertex  where γ (  ) > d eg + (  ) , then G does not have a compatible circuit. Ex. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Simple necessary condition Let γ (  ) be the size of the largest color class incident to  . Prop. If there exists a vertex  where γ (  ) > d eg + (  ) , then G does not have a compatible circuit. Ex. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Undirected eulerian graphs Thm. [Kotzig 1968] If G is a colored eulerian undirected graph and γ (  ) ≤ d eg (  ) / 2 for all vertices  , then G has a compatible circuit. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Undirected eulerian graphs Thm. [Kotzig 1968] If G is a colored eulerian undirected graph and γ (  ) ≤ d eg (  ) / 2 for all vertices  , then G has a compatible circuit. A colored eulerian digraph with γ (  ) ≤ deg + (  ) does not necessarily have a compatible circuit. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Undirected eulerian graphs Thm. [Kotzig 1968] If G is a colored eulerian undirected graph and γ (  ) ≤ d eg (  ) / 2 for all vertices  , then G has a compatible circuit. A colored eulerian digraph with γ (  ) ≤ deg + (  ) does not necessarily have a compatible circuit. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Splitting vertices We split vertices  where γ (  ) = d eg + (  ) . The graph G has a compatible circuit if and only if the graph G ′ after splitting has a compatible circuit. G ′ G  1   2 James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Splitting vertices We split vertices  where γ (  ) = d eg + (  ) . The graph G has a compatible circuit if and only if the graph G ′ after splitting has a compatible circuit. G ′ G James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Necessary Conditions Splitting vertices We split vertices  where γ (  ) = d eg + (  ) . The graph G has a compatible circuit if and only if the graph G ′ after splitting has a compatible circuit. G ′ G Henceforth, we may assume that γ (  ) < deg + (  ) for all  . James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Fixable vertices Excursions Def. Let T be an eulerian circuit of G and  a vertex of G . An excursion in T is the walk between consecutive visits to  . The excursion graph L T (  ) tracks the entering and exiting edges of the excursions at  . L T (  ) G   James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Fixable vertices Excursions Def. Let T be an eulerian circuit of G and  a vertex of G . An excursion in T is the walk between consecutive visits to  . The excursion graph L T (  ) tracks the entering and exiting edges of the excursions at  . L T (  ) G   James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Fixable vertices Fixable vertices We want to remove monochromatic transitions of T at  by rearranging the excursions at  . L T (  ) G   Def. Let M be any matching between E + (  ) and E − (  ) , and let L M (  ) be the implied excursion graph. A vertex  is fixable if L M (  ) has a compatible circuit for any matching M between E + (  ) and E − (  ) . James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Fixable vertices Def. Let M be any matching between E + (  ) and E − (  ) , and let L M (  ) be the implied excursion graph. A vertex  is fixable if L M (  ) has a compatible circuit for any matching M between E + (  ) and E − (  ) . Prop. If every vertex is fixable, then G has a compatible circuit. Proof. Pick a (not necessarily compatible) eulerian circuit T of G . Iteratively fix fixable vertices. The resulting circuit is compatible. � James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Fixable vertices Prop. A vertex is fixable unless γ (  ) = d eg + (  ) − 1 and there are 2 color classes of size γ (  ) with both in and out edges, and the other two edges are one incoming and one outgoing. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Fixable vertices Prop. A vertex is fixable unless γ (  ) = d eg + (  ) − 1 and there are 2 color classes of size γ (  ) with both in and out edges, and the other two edges are one incoming and one outgoing. Ex. The excursion graph L M (  ) has no compatible circuit. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Non-fixable vertices Nonfixable vertices Let S be the set of vertices that are not fixable. Let S 3 be the subset of S with vertices of outdegree three. We will consider colored eulerian digraphs with no nonfixable vertices of outdegree three. James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Non-fixable vertices Splitting nonfixable vertices We form a new graph G S by splitting each of the nonfixable vertices into three new vertices. G S G  1   2  3 James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Non-fixable vertices Splitting nonfixable vertices We form a new graph G S by splitting each of the nonfixable vertices into three new vertices. G S G  1   2  3 A compatible circuit through  can insert  1 into  2 or  3 , but  2 and  3 can not be combined. Can we glue vertices so that the whole graph is connected? James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Non-fixable vertices Component graph  2 A G S G  1   3 B C  3   1 D  2 James Carraher (UNL) Finding compatible circuits in eulerian digraphs

Non-fixable vertices Component graph  2 A H G G S G  1   3 B A  B C    3 C D    1 D  2 The component graph H G has components of G S as vertices. For each  ∈ S , put an edge in H G between D 1 ∋  1 and D 2 ∋  2 and an edge between D 1 ∋  1 and D 3 ∋  3 . James Carraher (UNL) Finding compatible circuits in eulerian digraphs