Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

Markus Chimani (Osnabrück University)

Joint work withGuiseppe Di Battista (University Roma Tre)

Fabrizio Frati (University of Sydney)

Karsten Klein (Monash University)

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markus . chimani @ uni - osnabrueck . de C-Planarity @ Flat Embedded Graph Drawing 2014

Clustered Planarity (C-Planarity)

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C = (G,T)

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Complexity of c-planarity testing

c-connected: O(|V|) [Dahlhaus 98]non c-connected: P? NP? open for ~20 years…

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Complexity for special cases

Polynomial cases…• C-connected [Feng et al. 95], [Dahlhaus 98]

• Completely c-connected [Cornelsen, Wagner 06]

• Almost c-connected [Gutwenger et al. 02]

• Extrovert [Goodrich et al. 05]

• “Cycles of Clusters” [Cortese et al. 05]

• and many more…

Unfortunately, most cases are very restricted and/or unnatural…Complexity even unknown for natural, seemingly simple subcases…• What if G is already embedded?• What if the cluster hierarchie is flat, i.e, only one level of clusters?• What if both?

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Now: Flat embedded clustered graphs• One level of clusters (= no nested clusters)• G is embedded.• Does there exist a drawing of the cluster regions such that the resulting

drawing is c-planar?

Known polynomial subcases:• Maximum face-size at most 5 or “single conflict” graph [Di Battista, Frati 09]• At most two components per cluster-induced graph [Jelinek et al. 09]• Clustered cycles:

3 clusters [Cortese et al. 05], 3 vertices/cluster [Jelinkova at al. 09]

Flat & Embedded

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Saturators

(Well known) idea: Flat embedded clustered graphs1. Each node starts within its own cluster region.2. Merge cluster regions along edges within the same cluster.

Step 2 is rather trivial to do in polynomial time…

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YObserveNo nodes of same cluster are adjacent after Step 2.

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Saturators

(Well known) idea: Flat embedded clustered graphs1. Each node starts within its own cluster region.2. Merge cluster regions along edges within the same cluster.3. Merge cluster regions of same cluster by adding

connectivity-establishing edges (con-edges) Saturator

Step 2 is rather trivial to do in polynomial time……but how to choose the con-edges in Step 3?

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Saturators




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AimFor each cluster, pick aspanning tree from itscon-edges, s.t. the differentspanning trees do not cross.

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Saturators




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AimFor each cluster, pick aspanning tree from itscon-edges, s.t. the differentspanning trees do not cross.

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Single-conflict graph

“Single conflict graph”If every con-edge is involved in at most one crossing, the problem is polynomial time solvable [Di Battista, Frati 09]

So, this is not the case here…

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2 vertices per cluster on each face

Our restriction:At most two vertices per cluster on each face.(“Single-conflict” graphs are a subset of those graphs)

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Not c-planar…

Here: By simple deduction, this graph is not c-planar

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Our algorithmAutomatically find a chain of deduction arguments to answer the c-planarity question.

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Algorithm – Overview

Sequence of 4 Tests (detect non-c-planarity) and8 Simplifications (shrink instance)

(1) if each remaining con-edge has at most one crossing thenreturn ALGORITHMFORSINGLECONFLICT [Di Battista, Frati 09]

(2) if Test1 = true then return “non-c-planar”(3) if Simpl1 applicable then perform Simpl1 & goto (1)(4) if Test2 = true then return “non-c-planar”(5) if Simpl2 applicable then perform Simpl2 & goto (1)(6) if Simpl3 applicable then perform Simpl3 & goto (1)(7–12) …(13) perform Simpl8 & goto (1) //always applicable of we got

that far

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Algorithm – Details

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ca

Let A[] the multigraph of all con-edges for cluster .

The first couple of tests & simplifications are trivial, e.g.• Test. Disconnected A[] No spanning tree non-c-planar

• Simplification. Bridge in A[] Merge vertices

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The next ones aren’t too hard either, e.g.• Test. Cyclic crossing sequence of odd length.

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The final ones are harder and more cumbersome – but interesting, e.g.• Simplifications 6–8. “-donut”

Simplification 6: If isomorphicconflicting structures at two spokes

remove one of the spokes and pick the crossing edges


“spokes”

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Algorithm

Wrapping upEventually (after several applications of the various simplifications),all -donuts vanish.Þ No con-edges with multiple crossings anymoreÞ Single-conflict graph solvable in polynomial time.

Since each test and simplification requires only polynomial time:

Theorem. The above algorithm decides c-planarity for flat embedded clustered graphs with at most two vertices per cluster on each facecorrectly in polynomial time.

Thank you

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The road ahead…

What’s the problem with more vertices of the same cluster on a common face?

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The road ahead…


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The road ahead…


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The road ahead…


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The road ahead…


Anyhow… is it possible to always deal with richer faces efficiently?

Thank you

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Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

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