Algorithms for Crossing Numbers

PackageInfo.g

@@ -391,6 +391,13 @@ Persons := [
     IsMaintainer  := false,
     Email         := "bspiers972@outlook.com"),
 
+  rec(
+    LastName      := "Toner",
+    FirstNames    := "Mark",
+    IsAuthor      := true,
+    IsMaintainer  := false,
+    Email         := "mark.toner@live.com"),
+
   rec(
     LastName      := "Tsalakou",
     FirstNames    := "Maria",

doc/crossingnumber.xml

@@ -0,0 +1,245 @@
+<#GAPDoc Label="DigraphCrossingNumberUpperBound">
+<ManSection>
+  <Attr Name="DigraphCrossingNumberUpperBound" Arg="digraph"/>
+  <Returns>A non-negative integer.</Returns>
+  <Description>
+    If <A>digraph</A> is a digraph, then <C>DigraphCrossingNumberUpperBound(<A>digraph</A>)</C>
+    returns the best known upper bound for the crossing number of <A>digraph</A>
+    <P/>
+
+    If <A>digraph</A> is planar it will return 0, if the digraph contains multiple parallel edges there
+    could be a lack of applicable theorems and the upper bound could become infinite. If the crossing
+    for <A>digraph</A> can be calculated this method will return that value.
+
+    <Example><![CDATA[
+gap> D := CompleteDigraph(5);;
+gap> DigraphCrossingNumberUpperBound(D);
+4
+gap> D := Digraph([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);
+<immutable multidigraph with 4 vertices, 11 edges>
+gap> DigraphCrossingNumberUpperBound(D);
+0
+gap> D := CompleteBipartiteDigraph(5, 4);;
+gap> DigraphCrossingNumberUpperBound(D);
+32]]></Example>
+</Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="DigraphCrossingNumberLowerBound">
+<ManSection>
+  <Attr Name="DigraphCrossingNumberUpperBound" Arg="digraph"/>
+  <Returns>A non-negative integer.</Returns>
+  <Description>
+    If <A>digraph</A> is a digraph, then <C>DigraphCrossingNumberLowerBound(<A>digraph</A>)</C>
+    returns the best known lower bound for the crossing number of <A>digraph</A>
+    <P/>
+
+    If <A>digraph</A> is planar it will return 0. If the crossing
+    for <A>digraph</A> can be calculated this method will return that value.
+
+    <Example><![CDATA[
+gap> D := CompleteDigraph(6);;
+gap> DigraphCrossingNumberLowerBound(D);
+12
+gap> D := Digraph([[1, 2, 4, 4, 5], [1, 3, 4, 5], [2, 1, 5], [1, 2, 4, 5], [1, 2]]);
+<immutable multidigraph with 5 vertices, 18 edges>
+gap> DigraphCrossingNumberLowerBound(D);
+0
+gap> D := CompleteBipartiteDigraph([5, 4, 5, 6]);;
+gap> DigraphCrossingNumberLowerBound(D);
+2282]]></Example>
+</Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="DigraphAddVertexCrossingPoint">
+<ManSection>
+  <Oper Name="DigraphAddVertexCrossingPoint" Arg="digraph, edge1, edge2"/>
+  <Returns>A <C>digraph</C> </Returns>
+  <Description>
+    If <A>digraph</A> is a digraph and <A>edge1,edge2</A> are edges in the
+    digraph, then adds a new vertex to <A>digraph</A> between <A>edge1</A> 
+    and <A>edge2</A>. Changes the edges of <A>digraph</A> by removing both 
+    <A>edge1</A> and <A>edge2</A> and adding an edge from <A>edge1[1]</A> 
+    to the new vertex and from the new vertex to <A>edge1[2]</A>, 
+    likewise for <A>edge[2]</A> for four new edges total. Returns the 
+    updated digraph if successful. Fails if <A>edge1</A> = <A>edge2</A>,
+    if any of the edges are loops, or if the edges are not present in the
+    <A>digraph</A>.
+    <Example><![CDATA[
+gap> D := CycleDigraph(4);;
+gap> DigraphAddVertexCrossingPoint(D, [1, 2], [2, 3]);
+<immutable digraph with 5 vertices, 6 edges>
+gap> D := CompleteDigraph(4);;
+gap> DigraphAddVertexCrossingPoint(D, [3, 5], [4, 3]);
+<immutable digraph with 7 vertices, 32 edges>
+]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="IsCubicDigraph">
+<ManSection>
+  <Prop Name="IsCubicDigraph" Arg="digraph"/>
+  <Returns><K>true</K> or <K>false</K>.</Returns>
+  <Description>
+    A <E>cubic digraph</E> is a digraph where each vertex has degree three.
+    Returns <K>true</K> if <A>digraph</A> is a cubic digraph and <K>false</K> if 
+    it is not. Will always return <K>false</K> if <A>digraph</A> has loops or
+    is a multidigraph.
+    <Example><![CDATA[
+gap> D := RandomTournament(4);
+<immutable tournament with 4 vertices>
+gap> IsCubicDigraph(D);
+true
+gap> D := Digraph([[2, 4], [3, 6], [4, 5], [], [1], [4, 5]]);
+<immutable digraph with 6 vertices, 9 edges>
+gap> IsCubicDigraph(D);
+true
+gap> D := CycleDigraph(7);
+<immutable cycle digraph with 7 vertices>
+gap> IsCubicDigraph(D);
+false
+gap> D := Digraph([[1, 1, 1]]);                   
+<immutable multidigraph with 1 vertex, 3 edges>
+gap> IsCubicDigraph(D);      
+false
+gap> D := CompleteMultipartiteDigraph([2, 3, 4, 1]);
+<immutable complete multipartite digraph with 10 vertices, 70 edges>
+gap> IsCubicDigraph(D);
+false]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="IsSemicompleteDigraph">
+<ManSection>
+  <Prop Name="IsSemicompleteDigraph" Arg="digraph"/>
+  <Returns><K>true</K> or <K>false</K>.</Returns>
+  <Description>
+    A <E>semicomplete digraph</E> is a digraph with at least on edge between
+    every pair of vertices. Returns <K>true</K> if <A>digrah</A> is semicomplete 
+    and <K>false</K> if it is not. Will always return <K>false</K> if 
+    <A>digraph</A> has loops or is a multidigraph. Returns <K>true</K> if
+    <A>digraph</A> is a tournament or complete digraph.
+    <Example><![CDATA[
+gap> D := RandomTournament(4);
+<immutable tournament with 4 vertices>
+gap> IsSemiComplete(D);
+true
+gap> D := Digraph([[2, 4], [3, 6], [4, 5], [], [1], [4, 5]]);
+<immutable digraph with 6 vertices, 9 edges>
+gap> IsSemicompleteDigraph(D);
+false
+gap> D := CompleteDigraph(7);
+<immutable complete digraph with 7 vertices>
+gap> IsCubicDigraph(D);
+true
+gap> D := Digraph([[1, 1, 1]]);                   
+<immutable multidigraph with 1 vertex, 3 edges>
+gap> IsSemicompleteDigraph(D);      
+false
+gap> D := CompleteMultipartiteDigraph([2, 3, 4, 1]);
+<immutable complete multipartite digraph with 10 vertices, 70 edges>
+gap> IsCubicDigraph(D);
+false]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="DigraphAllThreeCycles">
+<ManSection>
+  <Attr Name="DigraphAllThreeCircuits" Arg="digraph"/>
+  <Returns>A list of lists of three positive integers</Returns>
+  <Description>
+    Returns all possible three cycles in the digraph <A>digraph</A>. <P/>
+
+    A three cycle of a digraph is a directed cycle of length 3.
+
+    The returned list is sorted in increasing order of first vertex. 
+    Looping and repeated edges are not considered for creating possible
+    three cycles. 
+    <Example><![CDATA[
+gap> D := CompleteDigraph(4);;
+gap> DigraphAllThreeCircuits(D);
+[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 2 ], [ 1, 3, 4 ], [ 1, 4, 2 ], 
+  [ 1, 4, 3 ], [ 2, 3, 4 ], [ 2, 4, 3 ] ]
+gap> D := Digraph([[2], [3], [1]]);;
+gap> DigraphAllThreeCircuits(D);
+[ [ 1, 2, 3 ] ]
+gap> D := Digraph([[2, 4, 5], [1, 3], [1, 5], [5], [1, 4]]);;
+gap> DigraphAllThreeCircuits(D);
+[ [ 1, 2, 3 ], [ 1, 4, 5 ] ]
+gap> D := CompleteDigraph(3);;
+gap> DigraphAllThreeCircuits(D);
+[ [ 1, 2, 3 ], [ 1, 3, 2 ] ]
+gap> D := DigraphAddAllLoops(D);;
+gap> DigraphAllThreeCircuits(D);
+[ [ 1, 2, 3 ], [ 1, 3, 2 ] ]
+]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+
+<#GAPDoc Label="DigraphAllTriangles">
+<ManSection>
+  <Attr Name="DigraphAllTriangles" Arg="digraph"/>
+  <Returns>A list of lists of three positive integers</Returns>
+  <Description>
+    Returns all possible triangles in the digraph <A>digraph</A>. <P/>
+
+    A triangles of a digraph is an undirected cycle of length 3.
+
+    The returned list is sorted in lexicographic order on vertices. 
+    Looping and repeated edges are not considered for creating possible
+    triangless. 
+    <Example><![CDATA[
+gap> D := CompleteDigraph(4);;
+gap> DigraphAllTriangles(D);
+[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ] ]
+gap> D := Digraph([[2], [3], [1]]);;
+gap> DigraphAllTriangles(D);
+[ [ 1, 2, 3 ] ]
+gap> D := Digraph([[2, 4, 5], [1, 3], [1, 5], [5], [4]]);;
+gap> DigraphAllTriangles(D);
+[ [ 1, 2, 3 ], [ 1, 3, 5 ], [ 1, 4, 5 ] ]
+gap> D := CompleteDigraph(3);;
+gap> D := DigraphAddAllLoops(D);;
+gap> DigraphAllTriangles(D);
+[ [ 1, 2, 3 ] ]
+]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="DigraphLargePlanarSubdigraph">
+<ManSection>
+  <Attr Name="DigraphLargePlanarSubdigraph" Arg="digraph"/>
+  <Returns>A digraph.</Returns>
+  <Description>
+  An implementation of Algorithm A described by Calinescu et al. <Cite Key="CALINESCU1997"/>
+  which finds a large planar subgraph of a non-planar graph.
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="CompleteMultipartiteDigraphPartitionSize">
+<ManSection>
+  <Attr Name="CompleteMultipartiteDigraphPartitionSize" Arg="digraph"/>
+  <Returns>A list of positive integers.</Returns>
+  <Description>
+    Returns a list of the sizes of the partitions of a complete multipartite 
+    digraph <A>digraph</A>, the partitions are sorted in increasing order.
+    <P/>
+    <Example><![CDATA[
+gap> D := CompleteBipartiteDigraph(3, 4);;
+gap> CompleteMultipartiteDigraphPartitionSize(D);
+[ 3, 4 ]
+gap> D := CompleteMultipartiteDigraph([1, 5, 3]);  
+gap> CompleteMultipartiteDigraphPartitionSize(D);
+[ [ 1, 2, 3 ] ]]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>

doc/digraphs.bib

@@ -72,7 +72,7 @@ @article{Gab00
   Journal = {Information Processing Letters},
   Keywords = {Algorithms},
   Number = {34},
-  Pages = {107 - 114},
+  Pages = {107 - 114},:
   Title = {Path-based depth-first search for strong and biconnected
            components},
   Url = {https://www.sciencedirect.com/science/article/pii/S002001900000051X},
@@ -194,6 +194,20 @@ @InProceedings{US14
     isbn="978-3-319-11812-3"
 }
 
+@article{CA1997,
+title = {A Better Approximation Algorithm for Finding Planar Subgraphs},
+journal = {Journal of Algorithms},
+volume = {27},
+number = {2},
+pages = {269-302},
+year = {1998},
+issn = {0196-6774},
+doi = {https://doi.org/10.1006/jagm.1997.0920},
+url = {https://www.sciencedirect.com/science/article/pii/S0196677497909202},
+author = {Gruia Călinescu and Cristina G Fernandes and Ulrich Finkler and Howard Karloff},
+abstract = {The MAXIMUM PLANAR SUBGRAPH problem—given a graphG, find a largest planar subgraph ofG—has applications in circuit layout, facility layout, and graph drawing. No previous polynomial-time approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1/3, which is achieved simply by producing a spanning tree ofG. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4/9 instead of 1/3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.}
+}
+
 @article{Corneil1973,
     author = {Corneil, D. G. and Graham, B.},
     title = {An Algorithm for Determining the Chromatic Number of a Graph},