Edge-Deletion Ratio in Graphs

• Edge-deletion ratio is defined as a family of metrics that normalize the impact of removing edges to achieve specific graph properties.
• It measures trade-offs in edit-distance, spectral analysis, and epidemic control by balancing edge deletions against additions or invariant changes.
• Different formulations use denominators such as total edges, possible edits, or pre-deletion values to assess structural distance, intervention intensity, and approximation quality.

Edge-deletion ratio is a context-dependent normalization for the effect of removing edges. In hereditary edit-distance theory, it describes the deletions component within the minimum number of edge additions plus edge deletions needed to force a graph property; in graph-modification and epidemic-control problems, it is typically the fraction $k/|E|$ of edges that must be removed; in approximation algorithms, it can denote a multiplicative overhead normalized by a problem parameter; and in spectral graph theory, it can denote a post-deletion/pre-deletion quotient such as $E(G-e)/E(G)$ (Martin et al., 2020, Gaikwad et al., 2021, Kalantarzadeh et al., 10 Jul 2025, Mahato et al., 2021). This suggests that the term is best understood as a family of related ratios rather than a single invariant.

1. Terminological scope and normalizations

Across the literature, the phrase is attached to several distinct baselines. Some papers normalize by the total number of possible edges, some by the current edge set, some by the total number of edits, and some by a pre-deletion spectral quantity. The choice of denominator is not cosmetic: it determines whether the ratio measures structural distance, intervention intensity, approximation quality, or sensitivity of a graph invariant.

Setting	Quantity	Source
Hereditary edit distance	deletions over total edits, asymptotically $1$ or $0$ depending on the optimal strategy	(Martin et al., 2020)
Edge-deletion optimization	$r = k/|E(G)|$	(Gaikwad et al., 2021)
Tree-like classes	$\rho_\Pi(G)=k_\Pi(G)/|E|$	(Koch et al., 2022)
Local irregularity	$r(G)=I_e(G)/m$	(Bensmail et al., 18 Nov 2025)
DAG edge deletion	approximation factor $\alpha(k+1)$	(Kalantarzadeh et al., 10 Jul 2025)
Eccentricity energy	$$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$	(Mahato et al., 2021)
Bayesian networks	practical definitions include $r=|E_{\mathrm{del}}|/|E|$ and an evidence-weighted ratio	(Choi et al., 2012)

This multiplicity is explicit in the cited sources. In some cases, the ratio is part of the formal model; in others, it is introduced as a derived quantity that translates bounds on $E(G-e)/E(G)$0, energy, or KL divergence into a normalized scale.

2. Edit distance and deletion–addition tradeoffs

For hereditary graph properties, edge-deletion ratio appears inside the edit-distance function rather than as a standalone optimization parameter. For two $E(G-e)/E(G)$1-vertex graphs $E(G-e)/E(G)$2 on the same vertex set, the edit distance is

$E(G-e)/E(G)$3

and for a hereditary property $E(G-e)/E(G)$4 and target density $E(G-e)/E(G)$5,

$E(G-e)/E(G)$6

In the setting where $E(G-e)/E(G)$7 forbids an induced Erdős–Rényi random graph $E(G-e)/E(G)$8, the main theorem gives, a.a.s. as $E(G-e)/E(G)$9,

$1$0

for all $1$1 when $1$2, and for $1$3 for any $1$4 (Martin et al., 2020).

The minimum compares two pure strategies. The deletion-dominated strategy has normalized cost

$1$5

while the addition-dominated strategy has normalized cost

$1$6

Accordingly, if the deletion term is smaller, pure deletions are asymptotically optimal and the edge-deletion ratio, interpreted as deletions over total edits, is approximately $1$7. If the addition term is smaller, pure additions are optimal and the edge-deletion ratio is approximately $1$8. The equality threshold is

$1$9

At $0$0, both strategies have equal cost, and any mixture, including pure deletion or pure addition, achieves the asymptotic minimum (Martin et al., 2020).

This formulation makes the ratio a by-product of a two-strategy variational problem. It is not $0$1; instead it records which side of the edit budget—deletions or additions—carries the asymptotic optimum.

3. Minimum-deletion ratios in graph modification

In edge-deletion problems proper, the ratio is usually a direct normalization of the minimum deletion count. For $0$2-Free Edge Deletion, the decision problem is: $0$3 A natural measure is

$0$4

For the special case $0$5-Free Edge Deletion, the target condition is that every component of $0$6 has at most $0$7 vertices. The paper proves W[1]-hardness by treewidth, W[2]-hardness by solution size $0$8, feedback vertex set number, or pathwidth, FPT by vertex cover number, and a kernel with $0$9 vertices and $r = k/|E(G)|$0 edges; more precisely,

$r = k/|E(G)|$1

which implies, for reduced yes-instances,

$r = k/|E(G)|$2

The same paper gives an ILP formulation over twin classes induced by a vertex cover, so exact $r = k/|E(G)|$3, and therefore exact $r = k/|E(G)|$4, are computable in FPT time when the vertex cover number is small (Gaikwad et al., 2021).

For tree-like target classes, the same normalization appears as

$r = k/|E(G)|$5

where $r = k/|E(G)|$6 is the minimum number of deletions needed to reach the class $r = k/|E(G)|$7. Exact formulas are available in several cases. For forests,

$r = k/|E(G)|$8

and for connected inputs with target class “trees,”

$r = k/|E(G)|$9

Deletion to cacti is NP-complete even on bipartite inputs, but it becomes polynomial-time solvable on chordal inputs and admits a simpler and faster algorithm on quasi-threshold graphs (Koch et al., 2022).

A related normalization arises in local irregularity. The parameter

$\rho_\Pi(G)=k_\Pi(G)/|E|$0

induces

$\rho_\Pi(G)=k_\Pi(G)/|E|$1

The paper proves the conflict-based lower bound

$\rho_\Pi(G)=k_\Pi(G)/|E|$2

and proposes the conjecture

$\rho_\Pi(G)=k_\Pi(G)/|E|$3

for every connected graph $\rho_\Pi(G)=k_\Pi(G)/|E|$4. It verifies the conjecture for several families and gives exact or asymptotic ratios such as $\rho_\Pi(G)=k_\Pi(G)/|E|$5 if $\rho_\Pi(G)=k_\Pi(G)/|E|$6, $\rho_\Pi(G)=k_\Pi(G)/|E|$7, $\rho_\Pi(G)=k_\Pi(G)/|E|$8 for non-path trees, and

$\rho_\Pi(G)=k_\Pi(G)/|E|$9

for complete graphs (Bensmail et al., 18 Nov 2025).

4. Density caps and epidemic control

A second major use of the ratio normalizes deletions needed to bound subgraph density. In Bounded-Density Edge Deletion, with density

$r(G)=I_e(G)/m$0

the minimum deletion count is

$r(G)=I_e(G)/m$1

and the edge-deletion ratio is

$r(G)=I_e(G)/m$2

Exact formulas are available on several intervals: if $r(G)=I_e(G)/m$3, then $r(G)=I_e(G)/m$4 and $r(G)=I_e(G)/m$5; if $r(G)=I_e(G)/m$6, then

$r(G)=I_e(G)/m$7

if $r(G)=I_e(G)/m$8, then

$r(G)=I_e(G)/m$9

and if $\alpha(k+1)$0, then

$\alpha(k+1)$1

The problem is polynomial-time solvable on trees and cliques, NP-complete on planar bipartite graphs and split graphs, FPT with respect to vertex cover number, and W[1]-hard with respect to solution size and feedback edge number (Bazgan et al., 2024).

A complete complexity dichotomy is then given for $\alpha(k+1)$2-Bounded-Density Edge Deletion: if $\alpha(k+1)$3 or $\alpha(k+1)$4, the problem is polynomial-time solvable; otherwise it is NP-hard. For integral $\alpha(k+1)$5, the paper gives a randomized $\alpha(k+1)$6-time algorithm, and for fixed rational $\alpha(k+1)$7, it proves fixed-parameter tractability with respect to treewidth in time

$\alpha(k+1)$8

(Bentert et al., 6 Jan 2026).

In epidemic control, the same ratio becomes an intervention-intensity variable. For the SIS edge-deletion algorithm, with candidate set $\alpha(k+1)$9 and removed set $$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$0, the paper defines

$$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$1

The post-deletion linear upper-bounding matrix is

$$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$2

and if $$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$3, then $$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$4. In the homogeneous case $$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$5, $$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$6,

$$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$7

The greedy algorithm selects edges to maximize the marginal reduction of the supermodular upper bound

$$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$8

In the reported experiments, the Erdős–Rényi network used $$R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)$$9 deletions out of $r=|E_{\mathrm{del}}|/|E|$0, giving

$r=|E_{\mathrm{del}}|/|E|$1

while the Haslemere dataset used $r=|E_{\mathrm{del}}|/|E|$2 deletions out of $r=|E_{\mathrm{del}}|/|E|$3, giving

$r=|E_{\mathrm{del}}|/|E|$4

Both configurations converged to the disease-free equilibrium under the chosen parameters, and increasing $r=|E_{\mathrm{del}}|/|E|$5 reduced cumulative infections with diminishing returns (Hoang et al., 29 Jun 2026).

5. Approximation and probabilistic graphical models

In DAG Edge Deletion, the phrase is used in a different sense: as an approximation factor normalized by the path-length parameter. The problem asks for the minimum weight set of edges whose deletion leaves no directed path of length $r=|E_{\mathrm{del}}|/|E|$6. The paper defines the “Edge-Deletion Ratio” as the approximation factor $r=|E_{\mathrm{del}}|/|E|$7, where $r=|E_{\mathrm{del}}|/|E|$8 is determined by a randomized rounding analysis on an LP relaxation. The main bounds are: $r=|E_{\mathrm{del}}|/|E|$9 for i.i.d. uniform labels,

$E(G-e)/E(G)$00

for a refined independent label distribution, a barrier showing that no independent distribution can improve the analysis below

$E(G-e)/E(G)$01

and a

$E(G-e)/E(G)$02

approximation for bipartite DAGs and for instances with structured LP solutions (Kalantarzadeh et al., 10 Jul 2025).

A probabilistic-graphical interpretation appears in Bayesian network approximation by edge deletion. The paper does not explicitly define an edge-deletion ratio; instead, it gives evidence-based edge deletion by removing an edge $E(G-e)/E(G)$03 and replacing the CPT of $E(G-e)/E(G)$04 by

$E(G-e)/E(G)$05

A practical definition aligned with the method is the global ratio

$E(G-e)/E(G)$06

the per-node incoming deletion ratio

$E(G-e)/E(G)$07

and the evidence-weighted ratio

$E(G-e)/E(G)$08

The key KL-divergence bound for a single deleted edge is

$E(G-e)/E(G)$09

and for multiple deleted edges, under the stated restriction of at most one incoming deleted edge per node,

$E(G-e)/E(G)$10

Appendix B shows that the evidence term can be unbounded even when $E(G-e)/E(G)$11, so a small entropy-based ratio does not by itself guarantee a small approximation error (Choi et al., 2012).

6. Post-deletion spectral and energy ratios

A further usage compares a graph invariant after deleting an edge to its pre-deletion value. For the eccentricity matrix $E(G-e)/E(G)$12, the paper defines

$E(G-e)/E(G)$13

For the complete $E(G-e)/E(G)$14-partite graph $E(G-e)/E(G)$15 with $E(G-e)/E(G)$16 and $E(G-e)/E(G)$17, the paper proves that for any edge $E(G-e)/E(G)$18,

$E(G-e)/E(G)$19

hence $E(G-e)/E(G)$20. In the complete bipartite case, examples include

$E(G-e)/E(G)$21

The same paper also gives examples where eccentricity energy and distance energy move in opposite directions under the same deletion (Mahato et al., 2021).

For degree-based weighted adjacency matrices $E(G-e)/E(G)$22, the edge-deletion ratio can likewise be the post-/pre-deletion quotient of weighted energy or spectral radius. For complete graphs $E(G-e)/E(G)$23, with

$E(G-e)/E(G)$24

the exact formulas are

$E(G-e)/E(G)$25

The paper proves the sufficient condition that, for $E(G-e)/E(G)$26 and $E(G-e)/E(G)$27,

$E(G-e)/E(G)$28

whenever

$E(G-e)/E(G)$29

It also shows that ISI energy on regular tripartite graphs can increase after deletion: for $E(G-e)/E(G)$30,

$E(G-e)/E(G)$31

so the ratio is approximately $E(G-e)/E(G)$32. This corrects earlier claims about monotonic decrease and establishes that complete graphs and regular multipartite graphs behave differently under the same weighting (Rather et al., 9 Mar 2026).

7. Comparative structure and recurring themes

The surveyed uses of edge-deletion ratio differ along three axes. First, the denominator may be $E(G-e)/E(G)$33, $E(G-e)/E(G)$34, total edits, LP optimum, or a pre-deletion spectral quantity. Second, the numerator may count deleted edges, deleted edits within a mixed edit budget, or post-deletion loss or gain of a graph invariant. Third, the governing threshold may be combinatorial, probabilistic, or spectral: $E(G-e)/E(G)$35 in hereditary edit distance (Martin et al., 2020), the half-integral and $E(G-e)/E(G)$36 boundary in bounded-density deletion (Bentert et al., 6 Jan 2026), and $E(G-e)/E(G)$37 or $E(G-e)/E(G)$38 in SIS suppression (Hoang et al., 29 Jun 2026).

This suggests a useful classification. In modification problems, the ratio is usually a normalized optimum $E(G-e)/E(G)$39. In edit-distance theory, it measures the deletions share of a mixed add–delete optimum. In approximation algorithms, it is a guarantee relative to a relaxation. In spectral and matrix settings, it is a sensitivity quotient. The term is therefore stable only locally: its meaning is fixed by the optimization objective, the admissible operations, and the baseline against which deletion is normalized.