Edge-Irregulators of Graphs

• Edge-irregulators of graphs are sets of edges whose deletion makes each connected pair of vertices have distinct degrees.
• They serve as a measure of a graph’s distance from local irregularity with concrete bounds and exact values established for paths, cycles, and complete graphs.
• Algorithmic studies show that determining optimal edge-irregulators is NP-complete and W[1]-hard, yet fixed-parameter algorithms exist under specific structural parameters.

An edge-irregulator of a graph is a set of edges whose deletion produces a locally irregular graph—that is, a graph in which every edge joins vertices of distinct degree. Edge-irregulators provide an extremal and algorithmic measure of a graph’s distance to local irregularity and underpin a family of combinatorial optimization problems with intricate complexity and structural features. These concepts connect to classical topics in irregularity measures, decomposition, edit distance to regular graphs, graph labelings, and spectral bounds.

1. Fundamental Definitions

Given a finite, simple, undirected graph $G=(V,E)$, $G$ is locally irregular if for every edge $uv\in E$, $d_G(u)\neq d_G(v)$. An edge-irregulator is a subset $S\subseteq E(G)$ such that the subgraph $G-S$ is locally irregular. The minimal such $S$ defines the edge-irregulator number: $$_e(G) = \min\{\,|S|\,:\, S \subseteq E,\ G-S\ \text{is locally irregular}\,\}.$$ The computational problem, Optimal Edge-Irregulator, is:

• Input: Graph $G=(V,E)$ and integer $k$.
• Question: Does there exist $S\subseteq E$ with $|S|\leq k$ such that $G-S$ is locally irregular? (Fioravantes et al., 2023)

Another central parameter is the function $Ie(G)$, sometimes used interchangeably, measuring the minimum number of edges needing deletion for $G$ to become locally irregular (Bensmail et al., 18 Nov 2025).

2. Structural Properties and Extremal Bounds

Several structural bounds and extremal results have been established for $Ie(G)$:

• General Bound: For every connected graph $G$ with $m=|E|$, it is conjectured that $Ie(G)\leq m/3 + c$ for some absolute constant $c$ (the 1/3-conjecture), supported by constructions for paths, cycles, trees, and certain dense graphs (Bensmail et al., 18 Nov 2025).
• Sharpness: The conjecture is tight up to the value of $c$, as cycles of length $3k+2$ satisfy $Ie(C_{3k+2}) = \lfloor m/3 \rfloor + 1$.
• Exact Values: For paths $P_n$, $Ie(P_n) = \lfloor (n-1)/3 \rfloor$ or $\lceil (n-1)/3 \rceil$. For complete graphs $K_n$ with $n = t_k = k(k+1)/2$, $Ie(K_n) = |E(K_n)| - m_k$ where $m_k = k(k+1)(k-1)(3k+2)/24$ is the maximal size of a locally irregular subgraph (Bensmail et al., 18 Nov 2025).

A tabular summary of established values is provided below.

Graph Family	$Ie(G)$ or Bound
Path $P_n$	$\lfloor (n-1)/3 \rfloor$ or $\lceil (n-1)/3 \rceil$
Cycle $C_n$	$Ie(P_n)+1$
Complete bipartite	0 if $n\neq m$; $n$ if $n=m$
Subdivision-2 of $H$	$\geq |E(H)| = m/3$
Tree (non-path)	$\leq |E|/3$
$K_n$ ($n=t_k$)	$|E(K_n)| - m_k$

These results exploit parity, degree multisets, and combinatorial constructions, indicating that local irregularity imposes strong structural constraints.

3. Algorithmic and Parameterized Complexity Results

The edge-irregulator problem exhibits a multiform complexity landscape (Fioravantes et al., 2023, Bensmail et al., 18 Nov 2025):

• FPT (Fixed-Parameter Tractability):
  • Vertex Integrity, Cluster-Deletion Number, Neighborhood Diversity: There exist FPT algorithms parameterized by these quantities, using decomposition and Lenstra-style integer linear programming techniques. For vertex integrity $k$, the problem can be solved in $f(k)n^{O(1)}$ time (Fioravantes et al., 2023).
  • Solution Size + Maximum Degree, Vertex Cover Number: For parameterization by $k+\Delta$, kernelization yields a bounded-size kernel, and for vertex cover number $\tau$, dynamic programming with $O(2^{\tau^4}n^{O(1)})$ time is possible (Bensmail et al., 18 Nov 2025).
• Hardness Results:
  • NP-completeness: The problem is NP-complete even for planar bipartite graphs of maximum degree 6.
  • W[1]-hardness: For parameterization by solution size $k$, the feedback vertex set number, or treedepth (Fioravantes et al., 2023). This excludes efficient FPT algorithms for these parameters unless FPT=W[1]. As a consequence, no polynomial kernels are known.
• Lower Bounds: A conflict-based lower bound holds: If $E_C$ is the set of edges with $d(u)=d(v)$, then $Ie(G)\geq |E_C|/(2\Delta-1)$ (Bensmail et al., 18 Nov 2025).
• Algorithmic Table: (abbreviated for essential cases)

| Parameter | Edge-Irregulator Complexity | |------------------------------|-------------------------------------| | Solution size $k$ | W[1]-hard | | Feedback vertex set number | W[1]-hard | | Treedepth | W[1]-hard | | Vertex integrity | FPT (ILP-based) | | Cluster-deletion number | FPT (via integrity) | | Neighborhood diversity | FPT (small twin-classes) | | Planar + bipartite + $\Delta\leq6$ | NP-hard |

4. Variations and Related Graph Irregularity Notions

Edge-irregulators link to a broader set of irregularity concepts:

• Total Edge Irregularity Strength: Given a labeling $f:V\cup E\to\{1,\dots,k\}$, the smallest $k$ such that all $e\in E$ have distinct weights $wt_f(e)=f(e)+f(u)+f(v)$ is called the total edge irregularity strength, denoted $\mathrm{tes}(G)$ (Irwansyah et al., 2017, Pfender, 2010). For large graphs with bounded maximum degree, $\mathrm{tes}(G)=\lceil (m+2)/3\rceil$.
• Edit Distance to Regularity: The minimum number of edge edits (additions and deletions) to obtain a disjoint union of regular graphs is controlled, up to factorial in $\Delta$, by the number $i(G)$ of “separation vertices” (those adjacent to higher-degree neighbors) (Zeng, 2023).
• Locally Irregular Edge-Colorings: Partitioning $E(G)$ into color classes such that each induces a locally irregular subgraph, with the minimum number of colors being the irregular chromatic index $\chi’_{irr}(G)$. For subcubic graphs, $\chi’_{irr}(G)\leq4$, with many subclasses (e.g., claw-free, cycle permutation graphs) requiring $\leq3$ (Lužar et al., 2022).
• Edge-Irregulators under Walk Constraints: Instead of edge deletions, one may allow edge-multisets forming walks whose addition turns $G$ into a locally irregular multigraph. The minimum length of such an irregularising walk, $\mathsf{mlw}(G)$, has been studied with explicit bounds and exact values for families such as cycles, complete graphs, and trees (Bensmail et al., 26 Jun 2025).

5. Extremal Constructions, Proof Principles, and Kernelization Techniques

Key elements underlying both upper and lower bounds are:

• Subdivisions and Conflicts: Subdividing edges increases the lower bound for $Ie(G)$: if $G$ is obtained by subdividing every edge of $H$ twice, $Ie(G)\geq |E(H)|=|E(G)|/3$ (Bensmail et al., 18 Nov 2025).
• Integral LP Formulations: For FPT by vertex integrity, the problem reduces to an integer linear program with a parameter-dependent number of variables, executable via Lenstra’s algorithm (Fioravantes et al., 2023).
• Branching and Distance-to-Conflict Arguments: For FPT by solution size plus maximum degree, kernelization relies on the fact that every edge in a solution lies within bounded distance of a “conflicting” edge ($d(u)=d(v)$), allowing reduction to a small core (Bensmail et al., 18 Nov 2025).
• Reduction Gadgets: Hardness results employ reductions starting from canonical $k$-clique or general factor problems, using bipartite gadgets to encode instance-specific constraints (Fioravantes et al., 2023).

6. Connections to Regularizability and Related Edit Problems

The edge-irregulator problem relates closely to classical edit and labeling parameters:

• Regularizability: A graph is regularizable (via edge weight assignment) under different positivity constraints (arbitrary, non-negative, positive), and the minimal subgraphs that block regularizability (minimal "edge-irregulators" in this sense) coincide with certain unbalanced bipartite graphs or bridges (Franceschet et al., 2016).
• Spectral Irregularity Measures: The total irregularity $I(G)=\sum_{e\in E}|d(u)-d(v)|$ is bounded by Laplacian spectral data: $I(G)\leq\sqrt{m(nZ_G-4m^2)\lambda_{max}/n}$, linking combinatorial and spectral approaches (Goldberg, 2013).

7. Open Problems, Conjectures, and Future Directions

Principal conjectures include:

• 1/3-Conjecture: $Ie(G)\leq m/3 + c$ for all connected $G$ with some absolute $c$ (Bensmail et al., 18 Nov 2025).
• Complexity of FPT by Solution Size: It remains open whether the problem can be solved in FPT time with respect to solution size alone, as current lower bounds suggest otherwise.
• Edit Distance Tightness: Determining whether the upper bound on edit distance to a union of regular graphs, governed by separation-vertex count $i(G)$, is tight for all bounded-degree graphs (Zeng, 2023).
• Irregular Chromatic Index: Whether $\chi’_{irr}(G)\leq3$ for all decomposable graphs, except for a single cactus graph, is an open conjecture (Lužar et al., 2022).

These open directions demonstrate that edge-irregulators remain a central focus in the fine-grained analysis of local irregularity, graph labelings, and structural edit problems, with deep algorithmic, extremal, and spectral interconnections across contemporary graph theory.