A Polynomial Kernel for Face Cover on Non-Embedded Planar Graphs

TL;DR

Face Cover Number asks whether terminals lie on the boundaries of at most $k$ faces in some embedding. The paper introduces a bottom-up SPR-tree based dynamic programming framework and the notion of nice kernels to compress subgraphs while preserving face-interaction, achieving a cubic kernel of size $O(k^3)$ for planar graphs without a fixed embedding. This extends known kernel results from plane graphs to general planar graphs and delivers the first non-embedded kernel for Face Cover Number by carefully handling R-, S-, and P-nodes with gadget replacements and interface-preserving reductions. The techniques rely on a mix of embedded and non-embedded analysis, including rigidization to regain 3-connectivity and a sequence of kernel-preserving reductions, illustrating a path toward kernelization of topological problems on planar graphs. Open questions remain about the possibility of a linear kernel in the embedded setting and subexponential algorithms for the non-embedded case.

Abstract

Given a planar graph, a subset of its vertices called terminals, and $k \in \mathbb{N}$, the Face Cover Number problem asks whether the terminals lie on the boundaries of at most $k$ faces of some embedding of the input graph. When a plane graph is given in the input, the problem is known to have a polynomial kernel~\cite{GarneroST17}. In this paper, we present the first polynomial kernel for Face Cover Number when the input is a planar graph (without a fixed embedding). Our approach overcomes the challenge of not having a predefined set of face boundaries by building a kernel bottom-up on an SPR-tree while preserving the essential properties of the face cover along the way.

Figures (6)

• Figure 1: (left) A skeleton in black and orange with the orange dashed edge $xy$ being a virtual edge to a child node. The graph that replaces this virtual edge is blue. Terminals are red squares. The blue graph has face cover number one (all its terminals lie on its rectangularly drawn face). Hence, a kernel for the blue graph could in principle be a triangle on $x$, $y$ and a terminal (this also has face cover number one). This would even still contain the interface ($x,y$) between the blue and black graph. (right) However, if we replace the blue graph with its possible triangular kernel, the face cover of the entire graph drops from two to one.
• Figure 2: The gadget for semi-problematic virtual components. Red squares are terminals. Note that if $a$ and $b$ are fixed to be on the outer face, then the face boundaries are unique.
• Figure 3: Exemplary replacements made in \ref{['rr:unproblematic']} (top) and \ref{['rr:semi-problematic']} (bottom). The virtual edges are shown as orange dashed curves, their corresponding virtual components are shown in blue in the left figures and their corresponding replacements are shown on the right.
• Figure 4: Example of the result of boring edge removal (right) applied to a graph (left) in which terminals are red squares, and virtual edges and their endpoints are orange (the thick one being the corner edge).
• Figure 5: Example of the result of private face merging around the left terminal (left) and the result of exhaustive private face merging (right) applied to the graph from \ref{['fig:Rnode-ber']}.

Theorems & Definitions (34)

• Theorem 1
• Lemma 2: Hopcroft and Tarjan HopcroftTarjan73
• Lemma 3: Hopcroft and Tarjan HopcroftT73
• Lemma 3
• Definition 4: Types of virtual components
• Lemma 4
• Lemma 4
• Definition 5
• Lemma 6
• Lemma 6