Dynamic parameterized problems on unit disk graphs

TL;DR

The first data structures for fundamental parameterized problems on dynamic unit disk graphs are presented, and the goal is to maintain data structures so that the aforementioned parameterized problems on the unit disk graph induced by V can be solved efficiently.

Abstract

In this paper, we study fundamental parameterized problems such as $k$-Path/Cycle, Vertex Cover, Triangle Hitting Set, Feedback Vertex Set, and Cycle Packing for dynamic unit disk graphs. Given a vertex set $V$ changing dynamically under vertex insertions and deletions, our goal is to maintain data structures so that the aforementioned parameterized problems on the unit disk graph induced by $V$ can be solved efficiently. Although dynamic parameterized problems on general graphs have been studied extensively, no previous work focuses on unit disk graphs. In this paper, we present the first data structures for fundamental parameterized problems on dynamic unit disk graphs. More specifically, our data structure supports $2^{O(\sqrt{k})}$ update time and $O(k)$ query time for $k$-Path/Cycle. For the other problems, our data structures support $O(\log n)$ update time and $2^{O(\sqrt{k})}$ query time, where $k$ denotes the output size.

Figures (13)

• Figure 1: (a) The gray and red grid cells are $2$-neighboring cells of the red grid cell. (b) Illustration of $\textsf{Link}(u,v)$. (c) Illustration of $\textsf{Cut}(v)$.
• Figure 2: (a) Illustration of $\textsf{UD}(V)$. The gray part represents a 5-neighboring cell of the grid cells containing at least two vertices. These vertices are colored in red. Square vertices are vertices of $V\setminus V'$. (b) Illustration of $\textsf{UD}(V')$.
• Figure 3: (a) Illustration of $\textsf{UD}(V)$. The gray part represents the cells of $\boxplus_\textsf{core}$. Vertices of $V_\textsf{tri}$ are colored in red and square vertices are vertices of $V \setminus V_\textsf{core}$. (b) Illustration of $\textsf{UD}(V_\textsf{core})$.
• Figure 4: (a) Illustration of $\textsf{UD}(V)$. The vertices of $\textsf{UD}(V)$ in $V_\textsf{core}$, the boundary vertices, and the non-boundary vertices in $M$ are marked by the black, red, and blue vertices, respectively. The removed vertices and the contracted vertices in the construction of $M$ are marked by cross vertices and boxes, respectively. (b) Illustration of $M$. (c) In the construction of $M$, we are left with the induced path between $u$ and $v$, which will be contracted to the red edge in $M$. The two end edges of the path compose the bridge set of $T$.
• Figure 5: (a) A tree with 3 bridges $e_1$, $e_2$ and $e_3$. The vertex $w$ is the lowest common ancestor of the endpoints of $e_2$ and $e_3$. (b) The vertex $w$ has degree 3 after removing and contracting process, so it needs to appear in $M$.

Theorems & Definitions (26)

• Lemma 1
• Lemma 2: Theorem 1 in zehavi2021eth
• Lemma 3
• Lemma 4
• Theorem 5
• Lemma 6
• Theorem 7
• Lemma 8
• Theorem 11
• Lemma 12