Parameterised distance to local irregularity

TL;DR

The article analyzes the parameterised complexity of making a graph locally irregular by deleting a minimum set of vertices (vertex-irregulators) or edges (edge-irregulators). It delivers FPT algorithms for vertex-irregulators under neighborhood diversity, vertex integrity, and cluster deletion number, while proving W[1]-hardness for treedepth and the feedback vertex set. For edge-irregulators, it establishes NP-hardness even on restricted graph classes and W[1]-hardness with respect to the solution size, but also provides an FPT algorithm when parameterised by vertex integrity. Together, these results map the tractability landscape of both problems across common graph-structural parameters and set the stage for further study on approximability and related variants.

Abstract

A graph $G$ is \emph{locally irregular} if no two of its adjacent vertices have the same degree. In [Fioravantes et al. Complexity of finding maximum locally irregular induced subgraph. {\it SWAT}, 2022], the authors introduced and studied the problem of finding a locally irregular induced subgraph of a given a graph $G$ of maximum order, or, equivalently, computing a subset $S$ of $V(G)$ of minimum order, whose deletion from $G$ results in a locally irregular graph; $S$ is denoted as an \emph{optimal vertex-irregulator of $G$}. In this work we provide an in-depth analysis of the parameterised complexity of computing an optimal vertex-irregulator of a given graph $G$. Moreover, we introduce and study a variation of this problem, where $S$ is a substet of the edges of $G$; in this case, $S$ is denoted as an \emph{optimal edge-irregulator of $G$}. In particular, we prove that computing an optimal vertex-irregulator of a graph $G$ is in FPT when parameterised by the vertex integrity, neighborhood diversity or cluster deletion number of $G$, while it is $W[1]$-hard when parameterised by the feedback vertex set number or the treedepth of $G$. In the case of computing an optimal edge-irregulator of a graph $G$, we prove that this problem is in FPT when parameterised by the vertex integrity of $G$, while it is NP-hard even if $G$ is a planar bipartite graph of maximum degree $4$, and $W[1]$-hard when parameterised by the size of the solution, the feedback vertex set or the treedepth of $G$. Our results paint a comprehensive picture of the tractability of both problems studied here, considering most of the standard graph-structural parameters.

Figures (4)

• Figure 1: Overview of our results. A parameter $p$ appearing linked to a parameter $p'$ with $p$ being below $p'$ is to be understood as "there is a function $f$ such that, for each graph $G$, we have $p(G)\le f(p'(G))$". The bold font is used to indicate the parameters that we consider in this work. The asterisks are used to indicate that the corresponding result follows from observations based on the work in FMT22. In light blue (olive resp.) we exhibit the FPT results we provide for finding an optimal vertex (edge resp.) irregulator, denoted as ${\rm I}_v$ (${\rm I}_e$ resp.). In red we exhibit the ${\rm W}[1]$-hardness results we provide for both problems. The clique number of the graph is denoted by $\omega$.
• Figure 2: The construction in the proof of Theorem \ref{['thm:hardness']}. The dashed lines are used to represent the edges between the literal and the clause vertices.
• Figure 3: The gadget $H_b$, $b \in \mathbb{N}$, used in the proof of Theorem \ref{['thn:edges-w-hard-size-solution']}. The black vertices represent the vertices of the gadget. The white vertices represent either a set of the original vertices $V_i$, $i \in [k]$, or a set of edge vertices $U_{i,j}$, $1 \le i < j \le k$. In the construction, if $w$ is adjacent to vertices of a $V_i$, $i \in [k]$, then $b=|V_i|$ while if $w$ is adjacent to vertices of a $U_{i,j}$, $1 \le i < j \le k$, then $b=|U_{i,j}|$. In each copy of the gadget, the degrees of $w$ and $y$ are equal.
• Figure 4: The tree $T_{i,j}$ that is attached to the vertex $u_i$, where $j$ is such that $a_j\in \overline{L}(u_i)$, in the proof of Theorem \ref{['thm:edge-irr-w-hard']}. The value of $q$ is such that after attaching $T_{i,j}$ to $u_i$ (and thus including the edge $u_{i,j}u_i$) we have $d(u_{i,j})=2in^4-a'_j$.

Theorems & Definitions (19)

• Lemma 2.3
• proof
• Corollary 2.5
• Definition 3.1: La12
• Theorem 3.2
• proof
• Definition 3.3
• Theorem 3.4
• proof
• Definition 3.5: HKMN10