# Open Problems

###### Abstract

This document contains a list of open problems that Duc A. Hoang is interested in. It is also available in PDF format. Please get in touch if you have any idea/comment/question/update. You can find (probably more interesting) open problems from the following resources.

## 1 Preliminaries

Unless otherwise noted, all graphs are simple, undirected. We often denote by $P_{n},C_{n},K_{n},K_{m,n}$ the path, cycle, complete graph, complete bipartite graph on $n$ vertices, respectively. For two graphs $G,H$, the disjoint union of $G$ and $H$, denoted by $G+H$, is the graph with $V(G+H)=V(G)\cup V(H)$ and $E(G+H)=E(G)\cup E(H)$. For the terminology and notation not defined here, see the textbooks by Diestel  or West . For more details on Combinatorial Reconfiguration (https://en.wikipedia.org/wiki/Reconfiguration), see the surveys [13, 12, 11, 5] and the wiki page https://reconf.wikidot.com/.

## 2 Reconfiguration of Independent Sets under Token Sliding

### 2.1 Definitions

The double-broom graph $D_{r,n,s}$ is the tree obtained from a path $P_{n}=v_{1}v_{2}\dots v_{n}$ by attaching $r$ leaves to $v_{1}$ and $s$ leaves to $v_{n}$, where $r,n,s$ are positive integers. Two independent sets $I,J$ are adjacent under Token Sliding $(\mathsf{TS})$ if there exists $u,v\in V(G)$ such that $I\setminus J=\{u\}$, $J\setminus I=\{v\}$, and $uv\in E(G)$. The Token Sliding problem asks whether there is a $\mathsf{TS}$-sequence between $I$ and $J$ in $G$, that is, a sequence of independent sets starting with $I$ and ending with $J$ where any two consecutive members are adjacent under $\mathsf{TS}$. For a fixed positive integer $k$, the graph $\mathsf{TS}_{k}(G)$ contains all size-$k$ independent sets as its vertices/nodes, and two nodes are joined by an edge if their corresponding independent sets are adjacent under $\mathsf{TS}$. A graph $G$ is called a $\mathsf{TS}_{k}$-graph if there exists a graph $H$ such that $G$ and $\mathsf{TS}_{k}(H)$ are isomorphic.

### 2.2 Conjecture/Question

###### Open Problem 1.

What is the complexity of Token Sliding on outerplanar graphs? And more generally, on series-parallel graphs ($\equiv$ graphs of treewidth at most two)?

###### Open Problem 2.

Let $G$ be a graph. What are necessary and sufficient conditions for $G$ such that $\mathsf{TS}_{2}(G)$ is acyclic?

###### Conjecture 3.

Let $G$ be a forest. Then $\mathsf{TS}_{k}(G)$ is a forest if and only if $G$ is $\{2K_{2}+(k-2)K_{1},D_{2,2,2}+(k-2)K_{1},D_{2,4,2}+(k-3)K_{1}\}$-free, for some integer $k\geq 4$.

### 2.3 Background/Motivation

Independent Set Reconfiguration (ISR) is one of the most well-studied problems in the Combinatorial Reconfiguration [12, 5]. Among several variants of ISR, Token Sliding is of particular interest. One of the main reasons is that in Token Sliding one often needs to deal with the situation where a token must make “detours” by “moving away” (and then moving back later) to allow some other token to move . Alternatively, this can be seen as a kind of “bottleneck effect”  where a token may not be able to “reach” some vertices which are “far apart” from all tokens because some tokens “block the way”.

Our motivation for Open Problem 1 comes from the following two results that both appeared in 2014.

###### Theorem 4().

There exists a constant $c$ such that Token Sliding (and two other variants of ISR) is $\mathtt{PSPACE}$-complete on graphs of bandwidth at most $c$.

Indeed, since a graph of bandwidth at most $c$ also has pathwidth and treewidth at most $c$, Theorem 4 holds for graphs of pathwidth/treewidth at most $c$. Moreover, for $c=1$,

###### Theorem 5().

Token Sliding is in $\mathtt{P}$ on trees.

Naturally, on graphs of treewidth at most $c$, one may ask what the value of $c$ that separates “hard” from “easy” is. Toward answering this question, a first step is probably to consider $c=2$.

Our motivation for Open Problem 2 and Conjecture 3 comes from [1, 2]. In , the authors initiated the study of $\mathsf{TS}_{k}(G)$ from a purely graph-theoretic viewpoint. They continued their study in  focusing on those that are acyclic. Open Problem 2 involves characterizing general graphs, which seems to be quite challenging and a first step may be to consider well-known graph classes. Toward this direction, in , the authors proved a forbidden induced subgraph characterization of a tree $G$ satisfying that $\mathsf{TS}_{k}(G)$ is acyclic where $k\in\{2,3\}$ and Conjecture 3, if true, will provide a complete analysis of which trees having acyclic $\mathsf{TS}_{k}$-graphs.

## 3 Reconfiguring $k$-Path Vertex Covers on Trees under Token Sliding

### 3.1 Definitions

For a fixed integer $k\geq 2$, a $k$-path vertex cover of a graph $G$ is a vertex subset $I$ such that any path on $k$ vertices contains at least one member from $I$. Two $k$-path vertex covers are adjacent under Token Sliding ($\mathsf{TS}$) if there exists $u,v\in V(G)$ such that $I\setminus J=\{u\}$, $J\setminus I=\{v\}$, and $uv\in E(G)$. The $k$-Path Vertex Cover Reconfiguration under Token Sliding ($k$-PVCR-TS) problem asks if there is a $\mathsf{TS}$-sequence between two given $k$-path vertex covers $I$ and $J$ in $G$, that is, a sequence of $k$-path vertex covers starting with $I$ and ending with $J$ where any two consecutive members are adjacent under $\mathsf{TS}$.

### 3.2 Conjecture/Question

###### Open Problem 6.

What is the complexity of $k$-PVCR-TS on bipartite graphs? Or more restrictedly, on trees?

### 3.3 Background/Motivation

The $k$-PVCR-TS problem was first introduced in . The complexities of $k$-PVCR-TS and some other variants have been characterized in  for some graph classes, including planar and bounded bandwidth graphs, trees, paths, and cycles. However, $k$-PVCR-TS on trees remains open. Our motivation comes from an attempt in  to solve this problem for a subclass of trees called caterpillars (i.e., trees obtained by attaching leaves to a central path). Unlike the cases for independent sets, a token may have to make “detours” by “moving closer” (and then moving back later) to allow some other tokens to move.

## 4 Recoloring under Token Swapping

### 4.1 Definitions

For a fixed integer $k\geq 1$, a proper $k$-coloring $\alpha$ of a graph $G$ is a function $\alpha:V(G)\to\{1,2,\dots,k\}$ such that for any edge $uv\in E(G)$ we have $\alpha(u)\neq\alpha(v)$. If $G$ has a proper $k$-coloring, we say that it is $k$-colorable. Two proper $k$-colorings $\alpha,\beta$ are adjacent under Token Swapping if there exist $u,v\in V(G)$ such that $\alpha(u)=\beta(v)$, $\alpha(v)=\beta(u)$, $\alpha(w)=\beta(w)$ for any $w\in V(G)-\{u,v\}$, and $uv\in E(G)$. The $k$-Recoloring under Token Swapping ($k$-RTS) problem asks if there is a sequence of adjacent proper $k$-colorings of $G$ under Token Swapping between two given proper $k$-colorings $\alpha,\beta$. The graph $\mathcal{C}_{k}^{RTS}(G)$ takes all proper $k$-colorings of $G$ as its nodes and their adjacency are defined under Token Swapping.

### 4.2 Conjecture/Question

###### Open Problem 7.

What is the complexity of $k$-RTS on trees for some fixed $k\geq 3$?

###### Open Problem 8.

What is the smallest value of $k$ such that $\mathcal{C}_{k}^{RTS}(G)$ is connected for some given graph $G$?

###### Open Problem 9.

What are necessary and sufficient conditions for a $k$-colorable graph $G$ such that $\mathcal{C}_{k+1}^{RTS}(G)$ is connected?

### 4.3 Background/Motivation

The $k$-RTS problem lies between two of the most well-studied problems in Combinatorial Reconfiguration: Token Swapping  and Vertex Recoloring . (See [12, 11] for more details.) RTS was first proposed in  where the authors claimed that RTS remains $\mathtt{PSPACE}$-complete on planar graphs and can be solved in polynomial time on paths ($k=3$) and cographs (any $k$). Up to present, their results have not yet been published. The goal of our proposed open problems is to obtain a better understanding on the similarities and differences between $\mathcal{C}_{k}^{RTS}(G)$ and the well-stuided graph $\mathcal{C}_{k}(G)$—the graph whose nodes are proper $k$-colorings of $G$ and two nodes are adjacent if one can be obtained from the other by recoloring exactly one vertex.

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