On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

M. A. Bekos, S. Cornelsen, L. Grilli,S.-H. Hong, M. KaufmannOn the Recognition of Fan-Planar and Maximal Outer-Fan-Planar GraphsFan-Planar (FP) drawings introduced by Kaufmann and Ueckerdt, 2014

simple drawings

crossings are allowed, but only in the form of fan-crossingsFan-Planar (FP) drawingsfan-crossingif an edge e is crossed by two or more edges e1, , ene1, , en must share a common vertex (the apex of the fan) Forbidden crossing pattern Ia drawing is fan-planar iff it does not contain an edge crossed by two independent edgesForbidden crossing pattern IIa drawing is fan-planar iff it does not contain an edge e crossed by two edges having theircommon end-point on different sides of e NOT IMPORTANT for our purposesOuter-Fan-Planar (OFP) drawingsOFP drawing: FP drawing + all vertices on the outer-face outer edgeinner edgemixed edgein part inner andin part outer

Maximal OFP drawingsmaximal OFP drawing: OFP drawing s.t. no two disjoint vertices can be connected with a curve without violating OFPOFP drawing that is not maximal OFPMaximal OFP drawingsmaximal OFP drawing: OFP drawing s.t. no two disjoint vertices can be connected with a curve without violating OFPmaximal OFP drawingFP graphs and OFP graphsFP graph: a graph that admits an FP drawingat most 5n 10 edges, tight bound [Kaufmann and Ueckerdt, 2014]

OFP graph: a graph that admits an OFP drawingP FP (P planar)OFP FPP OFP

Maximal OFP graphsmaximal OFP graph: OFP graph that admits a maximal OFP drawingMaximal OFP graphsmaximal OFP graph: OFP graph that admits a maximal OFP drawing

CORRECTmaximal OFP graph: OFP graph s.t. the addition of any edge makes it no longer OFPany of its embedding is either maximal OFP or is not OFP

Maximal OFP 2-connectivitycut-vertexMaximal OFP 2-connectivitycut-vertexMaximal OFP 2-connectivitycut-vertexMaximal OFP 2-connectivitycut-vertexMaximal OFP Hamiltonicitya maximal OFP drawing of a maximal OFP graph GMaximal OFP Hamiltonicitya maximal OFP drawing of a maximal OFP graph GMaximal OFP Hamiltonicityvivi+1Maximal OFP circular drawingsCMaximal OFP circular drawingsCOur Resultsrecognition of maximal OFP graphs + computation of a maximal OFP embedding in the positive case3-connected graphs: linear time algorithm2-connected graphs: polynomial time algorithm based on SPQR-tree

NP-hard proof for the fan-planarity testing problem in the fixed rotation system setting

recognition of maximal OFP graphs3-connected case Outer, 2-hop and long edgesonce fixed the embedding (cyclic ordering) three kinds of edgesv1v2v4v6v8v7v3v5outer edge 2-hop edgelong edgeComplete 2-Hop (C2H) graphsC2H graph: a cycle C of n 4 vertices + all 2-hop edgescycle C induces a cyclic ordering of the verticesv5v10v7v3v2v6v4v1v9v83-connectedHamiltonian4-regular for n 52n edgesv8v6Complete 2-Hop (C2H) graphsdifferent embeddings (drawings) of a same C2H graphv5v10v7v3v2v6v4v1v9v8v3v9v7v2v10v5v1v4C1C2C1 is not OFPC2 is maximal OFPC2H embeddings with long edges: any drawing of a C2H graph with a long edge is not OFP

vjvi+1vi-1vivj-1vj+1maximal OFP consecutive vertices on theboundary of are adjacent(vi, vj) a long edgeC2H embeddings with long edges: any drawing of a C2H graph with a long edge is not OFP

3-connectivity and absence of inner vertices edge (vi+1, vj+1)vjvi+1vi-1vivj-1vj+1C2H embeddings with long edges: any drawing of a C2H graph with a long edge is not OFP

3-connectivity and absence of inner vertices edge (vi-1, vj-1)vjvi+1vi-1vivj-1vj+1C2H embeddings with long edges: any drawing of a C2H graph with a long edge is not FP

3-connectivity and absence of inner vertices edge (vi-1, vj-1)vjvi+1vi-1vivj-1vj+1 embeddings with long edges are not FPsince embeddings without long edges are maximal OFP C2H graphs are maximal OFPC2H graphs are maximal OFP graphsG: a C2H graphany embedding of G with long edges is not maximal OFPall the others are maximal OFP C2H graphs maximal OFP graphsHamiltonian3-cyclesn = 3C2H graphsn 4LED graphsLong Edge Decomposable graphsn 6 3-connected maximal OFP2-connected maximal OFPgraph G3-connected, n 3 verticesn = 3n 4outputis GLED-decomposable?is G a C2H?YesYesNoYes/Nomaximal OFP recognition for 3-connected graphs, algorithm structure any embedding long edge(s)Recognition of C2H graphsC2H graphs for n = 4, 5, 6C2Hnn#OFP embeddings4563all cyclic orderings12all cyclic orderingsis a Kn?

4C2H6 is easy to recognize G6 C2H6G6 is 4-regular(deg(v) = 4 v of G6) () trivial() G6 is 4-regularC2H6 is easy to recognize G6 C2H6G6 is 4-regular(deg(v) = 4 v of G6) () trivial() G6 is 4-regular G6 is a clique minus three pairwise disjoint edges: (u,v), (w,z), (x,y) K6uvwzxyC2H6 is easy to recognize G6 C2H6G6 is 4-regular(deg(v) = 4 v of G6) () trivial() G6 is 4-regular G6 is a clique minus three pairwise disjoint edges: (u,v), (w,z), (x,y) uvwzxyG6C2H6 is easy to recognize G6 C2H6G6 is 4-regular(deg(v) = 4 v of G6) () trivial() G6 is 4-regular G6 is a clique minus three pairwise disjoint edges: (u,v), (w,z), (x,y) consider cycle u w x v z y u uvwzxyG6C2H6 is easy to recognize G6 C2H6G6 is 4-regular(deg(v) = 4 v of G6) () trivial() G6 is 4-regular G6 is a clique minus three pairwise disjoint edges: (u,v), (w,z), (x,y) consider cycle u w x v z y u uvwzxyG6uwxvzyG6 C2H6C2Hn with n 7vNeighbors of v induce a pathvuwzwremove v from C2HnFrom C2Hn to C2Hn-2 with n 7 uwzwmerge w and w into a unique vertex wadd edge (u,z)From C2Hn to C2Hn-2 with n 7 uwzthe result is a C2H graph of n 2 verticesnote: there exists exactly 1 OFP embedding algorithmRecognitionC2H()Input: a 3-connected graph G, with n 4 verticesOutput: TRUE iff G C2Hn

if (n 5) return whether G is a clique;if (n == 6) return whether G is 4-regular;if (n 7 and G is 4-regular) { pick any vertex v of G; //let u-w-w-z be the path induced by its neighbors merge neighbors w and w; add edge (u,z); //let Gn-2 be the resulting graph return algorithmRecognitionC2H(Gn-2);}algorithmRecognitionC2H()Input: a 3-connected graph G, with n 4 verticesOutput: TRUE iff G C2Hn

if (n 5) return whether G is a clique;if (n == 6) return whether G is 4-regular;if (n 7 and G is 4-regular) { pick any vertex v of G; //let u-w-w-z be the path induced by its neighbors merge neighbors w and w; add edge (u,z); //let Gn-2 be the resulting graph return algorithmRecognitionC2H(Gn-2);}algorithmRecognitionC2H()Input: a 3-connected graph G, with n 4 verticesOutput: TRUE iff G C2Hn

if (n 5) return whether G is a clique;if (n == 6) return whether G is 4-regular;if (n 7 and G is 4-regular) { pick any vertex v of G; if (neighbors u,w,w,z of v do not induce path u-w-w-z) return FALSE; merge neighbors w and w; add edge (u,z); //let Gn-2 be the resulting graph return algorithmRecognitionC2H(Gn-2);}graph G3-connected, n 3 verticesn = 3n 4outputis GLED-decomposable?is G a C2H?YesYesNoYes/Nomaximal OFP recognition for 3-connected graphs, algorithm structure any embedding long edge(s)DONERecognition of LED graphsLong Edge Decomposable graphsMaximal OFP graphs C2H graphsG: a maximal OFP graph that is not C2H, with n 6 vertices any drawing of G has a long edge e = (u,v)uvMaximal OFP graphs C2H graphs with a long edge e = (u,v) K4 G s.t.vertices of K4 appear consecutively on the boundary of exactly one vertex w of K4 has degree 3 in G(K4 may contain end-vertices u,v of the long edge)uvMaximal OFP graphs C2H graphs with a long edge e = (u,v) K4 G s.t.vertices of K4 appear consecutively on the boundary of exactly one vertex w of K4 has degree 3 in G(K4 may contain end-vertices u,v of the long edge)uvExample 1wxzyK4deg(w) = 3Maximal OFP graphs C2H graphs with a long edge e = (u,v) K4 G s.t.vertices of K4 appear consecutively on the boundary of exactly one vertex w of K4 has degree 3 in G(K4 may contain end-vertices u,v of the long edge)uvExample 2wzK4deg(w) = 3G is a maximal OFP graph and is not C2Hif and only ifG {w} is maximal OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

Long Edge Decomposition (LED)ExampleLong Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

Examplew1Long Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

ExampleLong Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

Examplew2Long Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

ExampleLong Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

Examplew3Long Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

ExampleLong Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

Examplew4Long Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

ExampleLong Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

Examplew5 = wn-3Long Edge Decomposition (LED)G is a maximal OFP graph and is not C2Hif and only ifG {w} is OFP and not C2H test: check whether it is possible to repeat such a removal until a triangle is obtained

ExamplexyzFinding all maximal OFP embeddingsreinsert vertices w in reverse order, starting from a triangle x, y, zStep 1) reinsertion of wn-3 3 embeddingsStep 2) reinsertion of wn-4 6 embeddingsStep 3) reinsertion of wn-5 4 embeddingsStep 4) reinsertion of wn-i 4 embeddingsRecognition of 2-connected maximal OFP graphs Graph decompositionG: 2-connected maximal OFP{s,t}: separation pairstststGlGrGGraph decompositionG: 2-connected maximal OFP{s,t}: separation pairstststGlGrG G1, Gr: 2-connectedmaximal OFP withedge {s,t} in the outerfaceGraph decompositionG: 2-connected maximal OFP{s,t}: separation pairstststGlGrG G1, G2: 2-connectedmaximal OFP withedge {s,t} in the outerfaceGl,Gr maximal OFP G maximal OFPGl is maximal OFPGr is maximal OFPststGl,Gr maximal OFP G maximal OFPG is NOT maximal OFPst but edges of this kind can connect only neighbors of vertices {s,t}NP-hardness of FP-FRSFan-Planarity with Fixed-Rotation System (FP-FRS)Rotation systemrotation at a vertex: a cyclic ordering of its incident edgesrotation system: a list of rotations of all vertices same graph and rotation systembut different sets of edge crossingsFP-FRS: problem definitionInput: a pair G, Ri.e. a graph G = (V, E) with a fixed rotation system RQuestion: does G admit a fan-planar drawing that preserves rotation system R?

our result: FP-FRS is NP-hardreduction from 3-Partition (3P)

3P: problem definitionInput: a multi-set A = {a1, a2, , a3m} of 3m integers s.t. ai = mB ai (B/4, B/2) i = 1, 2, , 3m

Question: can A be partitioned into m subsets A1, A2, , Am, each of cardinality 3, whose elements sum to B?

i = 13m3P: problem definitionInput: a multi-set A = {a1, a2, , a3m} of 3m integers s.t. ai = mB ai (B/4, B/2) i = 1, 2, , 3m

Question: can A be partitioned into m subsets A1, A2, , Am, each of cardinality 3, whose elements sum to B?

Example for m = 3 and B = 24, (B/4, B/2) = (6,12)A = {7, 7, 7, 8, 8, 8, 8, 9, 10}is A a Yes-instance of 3P?

Yes!A1 = {7, 7, 10} A2 = {7, 8, 9} A3 = {8, 8, 8}i = 13mBarrier gadgetsC2HnBarrier gadgetsC2Hnfan-planarityviolationBarrier gadgetsC2HnBarrier gadgetsC2HnTransforming 3P into FP-FRS uv12jmTransversal paths j from u to v uv12jmTransversal paths j from u to v uv12jm81uvA = {7, 7, 7, 8, 8, 8, 8, 9, 10} m = 3,B = 24777888910K = B/2 + 1 = 13 8uvA = {7, 7, 7, 8, 8, 8, 8, 9, 10} m = 3,B = 24777888910K = B/2 + 1 = 13 81uvA = {7, 7, 7, 8, 8, 8, 8, 9, 10} m = 3,B = 24777888910K = B/2 + 1 = 13 813uvA = {7, 7, 7, 8, 8, 8, 8, 9, 10} m = 3,B = 24777888910K = B/2 + 1 = 13 813we assign to each path (3m 3)K + B uvA = {7, 7, 7, 8, 8, 8, 8, 9, 10} m = 3,B = 247778888910K = B/2 + 1 = 13 1231 A1 = {7,7,10}2 A2 = {7,8,9}3 A3 = {8,8,8}every path passes through exactly 3 central cells Conclusion and open problemsalgorithm for the recognition of maximal OFP graphs and for computing maximal OFP embeddings in the positive case3-connected case2-connected case

NP-hardness of FP-FRS

what about the recognition of maximal OFP graphs?The End