Exact Algorithms for Clustered Planarity with Linear Saturators

TL;DR

It is shown that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most $3$, thus excluding the algorithmic use of the number of clusters as a parameter.

Abstract

We study Clustered Planarity with Linear Saturators, which is the problem of augmenting an $n$-vertex planar graph whose vertices are partitioned into independent sets (called clusters) with paths - one for each cluster - that connect all the vertices in each cluster while maintaining planarity. We show that the problem can be solved in time $2^{O(n)}$ for both the variable and fixed embedding case. Moreover, we show that it can be solved in subexponential time $2^{O(\sqrt{n}\log n)}$ in the fixed embedding case if additionally the input graph is connected. The latter time complexity is tight under the Exponential-Time Hypothesis. We also show that $n$ can be replaced with the vertex cover number of the input graph by providing a linear (resp. polynomial) kernel for the variable-embedding (resp. fixed-embedding) case; these results contrast the NP-hardness of the problem on graphs of bounded treewidth (and even on trees). Finally, we complement known lower bounds for the problem by showing that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most $3$, thus excluding the algorithmic use of the number of clusters as a parameter.

Figures (2)

• Figure 1: (a) A partial clique planar representation focused on a cluster. (b) Canonical representation of the cluster in (a). A linear saturation of the vertices of the cluster corresponding to (b).
• Figure 2: (a) Example for \ref{['def:extractTriple']}: The paths in $\cal S$ are drawn as black curves, and the vertices in $U$ are marked by disks. We have $M=\{\{a,b\},\{c,e\},\{f,i\}\}, P=\{m,o,r\}, D=\{d,g,h,j,k,l,n,p,q\}$ and $U\setminus (V(M)\cup P\cup D)=\{s,t\}$. (b) Example for \ref{['def:complement']}: The vertices in $U$ are marked by disks, and their association with the clusters is indicated by colors (explicitly, the clusters are $\{\{a,b,p,q,r,s,t\},\{c,d,e\},\{f,g,h,i,l,m,n,o\},\{j,k\}\}$). Suppose $M_{\mathsf{in}}=\{\{a,b\},\{c,e\},\{f,i\}\}, P_{\mathsf{in}}=\{m,o,r\}, D_{\mathsf{in}}=\{d,g,h,j,k,l,n,p,q\}$ and $U\setminus (V(M_{\mathsf{in}})\cup P_{\mathsf{in}}\cup D_{\mathsf{in}})=\{s,t\}$, and $M_{\mathsf{out}}=\{\{f,o\},\{i,m\},\{r,a\}\}$, $P_{\mathsf{out}}=\{c,e\}$, $D_{\mathsf{out}}=\{s,t\}$ and $U\setminus (V(M_{\mathsf{out}})\cup P_{\mathsf{out}}\cup D_{\mathsf{out}})=\{b,d,g,h,j,k,l,n,p,q\}$. Then, the triples $T_{\mathsf{in}}=(M_{\mathsf{in}},P_{\mathsf{in}},D_{\mathsf{in}})$ and $T_{\mathsf{out}}=(M_{\mathsf{out}},P_{\mathsf{out}},D_{\mathsf{out}})$ are complementary, and $G_{T_{\mathsf{in}},T_{\mathsf{out}}}$ is a subgraph of the illustrated graph induced by $\{a,b,c,e,f,i,m,o,r\}$. The pendants are the endpoints of the edges going inside or outside the cycle (the vertex $b$, for example, is not adjacent to a pendant, while the vertex $c$ is).

Theorems & Definitions (42)

• Definition A: CPLS-Completion Problem
• Definition B: CPLSF-Completion Problem
• Theorem 1
• Definition 2: Non-Crossing Matchings and the Partition MatPenDel
• Definition 4: Partial Solution
• Definition 5: Extracting a Triple from a Partial Solution
• Definition 7: Complementary Triples in MatPenDel
• Definition 8: Partial Solution Compatible with $(T_\mathsf{in},I,E_\mathsf{in})$
• Definition 9: Sensibility of $(T_\mathsf{in},I,E_\mathsf{in})$
• Lemma 10