--- title: Edge-Deletion Ratio in Graphs url: https://www.emergentmind.com/topics/edge-deletion-ratio type: topic --- # Edge-Deletion Ratio in Graphs 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 \(E(G-e)/E(G)\) [2007.08409], [2102.06068], [2507.07943], [2107.03237]. 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 | [2007.08409] | | Edge-deletion optimization | \(r = k/|E(G)|\) | [2102.06068] | | Tree-like classes | \(\rho_\Pi(G)=k_\Pi(G)/|E|\) | [2210.03839] | | Local irregularity | \(r(G)=I_e(G)/m\) | [2511.14514] | | DAG edge deletion | approximation factor \(\alpha(k+1)\) | [2507.07943] | | Eccentricity energy | \(R_\varepsilon(G,e)=E_\varepsilon(G-e)/E_\varepsilon(G)\) | [2107.03237] | | Bayesian networks | practical definitions include \(r=|E_{\mathrm{del}}|/|E|\) and an evidence-weighted ratio | [1207.1370] | 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 \(k\), 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 \(n\)-vertex graphs \(G,H\) on the same vertex set, the edit distance is \[ \operatorname{dist}(G,H)=|E(G)\triangle E(H)|\binom{n}{2}^{-1}, \] and for a hereditary property \(\mathcal H\) and target density \(p\), \[ \mathrm{ed}_{\mathcal H}(p):=\limsup_{n\to\infty}\ \max_{\substack{|V(G)|=n\\ e(G)=\lfloor p\binom{n}{2}\rfloor}}\ \min\bigl\{\operatorname{dist}(G,H): H\in\mathcal H,\ V(H)=V(G)\bigr\}. \] In the setting where \(\mathcal H=(F)\) forbids an induced Erdős–Rényi random graph \(F\sim\mathbb G(n_0,p_0)\), the main theorem gives, a.a.s. as \(n_0\to\infty\), \[ \mathrm{ed}_{\mathcal H}(p) =(1+o(1))\,\frac{2\log n_0}{n_0}\cdot \min\left\{\frac{p}{-\log(1-p_0)},\ \frac{1-p}{-\log p_0}\right\}, \] for all \(p\in[0,1]\) when \(p_0\in[1-1/\varphi,1/\varphi]\), and for \(p\in[1/3,2/3]\) for any \(p_0\in(0,1)\) [2007.08409]. The minimum compares two pure strategies. The deletion-dominated strategy has normalized cost \[ \frac{2\log n_0}{n_0}\cdot \frac{p}{-\ln(1-p_0)}, \] while the addition-dominated strategy has normalized cost \[ \frac{2\log n_0}{n_0}\cdot \frac{1-p}{-\ln p_0}. \] 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 \(1\). If the addition term is smaller, pure additions are optimal and the edge-deletion ratio is approximately \(0\). The equality threshold is \[ p^*=\frac{\ln(1-p_0)}{\ln\bigl(p_0(1-p_0)\bigr)}. \] At \(p=p^*\), both strategies have equal cost, and any mixture, including pure deletion or pure addition, achieves the asymptotic minimum [2007.08409]. This formulation makes the ratio a by-product of a two-strategy variational problem. It is not \(k/|E|\); 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 \(\mathcal F\)-Free Edge Deletion, the decision problem is: \[ \textbf{Input: }G=(V,E),\ \mathcal{F},\ k\in \mathbb{Z}_{>0},\qquad \textbf{Question: } \exists E'\subseteq E(G),\ |E'|=k,\ \text{s.t. } G\setminus E' \text{ contains no } F\in\mathcal{F} \text{ as a subgraph?} \] A natural measure is \[ r=\frac{k}{|E(G)|}. \] For the special case \(\mathcal T_{h+1}\)-Free Edge Deletion, the target condition is that every component of \(G\setminus E'\) has at most \(h\) vertices. The paper proves W[1]-hardness by treewidth, W[2]-hardness by solution size \(k\), feedback vertex set number, or pathwidth, FPT by vertex cover number, and a kernel with \(O(hk)\) vertices and \(O(h^2k)\) edges; more precisely, \[ |V(G)|\le 2kh,\qquad |E(G)|\le 2kh^2+k, \] which implies, for reduced yes-instances, \[ r=\frac{k}{|E(G)|}\ge \frac{1}{2h^2+1}. \] The same paper gives an ILP formulation over twin classes induced by a vertex cover, so exact \(k\), and therefore exact \(r\), are computable in FPT time when the vertex cover number is small [2102.06068]. For tree-like target classes, the same normalization appears as \[ \rho_\Pi(G):=\frac{k_\Pi(G)}{|E|}, \] where \(k_\Pi(G)\) is the minimum number of deletions needed to reach the class \(\Pi\). Exact formulas are available in several cases. For forests, \[ k_{\mathrm{Forest}}(G)=|E|-(n-c),\qquad \rho_{\mathrm{Forest}}(G)=\frac{|E|-(n-c)}{|E|}, \] and for connected inputs with target class “trees,” \[ k_{\mathrm{Tree}}(G)=|E|-(n-1),\qquad \rho_{\mathrm{Tree}}(G)=\frac{|E|-(n-1)}{|E|}. \] 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 [2210.03839]. A related normalization arises in local irregularity. The parameter \[ I_e(G)=\min\big\{ k\ge 0:\exists F\subseteq E(G),\ |F|=k,\ \text{and}\ G-F\ \text{is locally irregular}\big\} \] induces \[ r(G)=\frac{I_e(G)}{m}. \] The paper proves the conflict-based lower bound \[ I_e(G)\ge \left\lceil \frac{\operatorname{Conf}(G)}{2\Delta-1}\right\rceil, \] and proposes the conjecture \[ I_e(G)\le \frac{m}{3}+c \] for every connected graph \(G\). It verifies the conjecture for several families and gives exact or asymptotic ratios such as \(r(K_{n,m})=0\) if \(n\neq m\), \(r(K_{n,n})=1/n\), \(I_e(T)\le |E(T)|/3\) for non-path trees, and \[ r(K_n)=\Theta\Big(\frac{1}{\sqrt n}\Big)\to 0 \] for complete graphs [2511.14514]. ## 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 \[ \rho(H)=\frac{|E(H)|}{|V(H)|},\qquad \rho^*(G)=\max_{H\subseteq G}\frac{|E(H)|}{|V(H)|}, \] the minimum deletion count is \[ k^*(G,\tau_\rho):=\min\{\,|E'|:E'\subseteq E,\ \rho^*(G-E')\le \tau_\rho\,\}, \] and the edge-deletion ratio is \[ r^*(G,\tau_\rho)=\frac{k^*(G,\tau_\rho)}{|E|}. \] Exact formulas are available on several intervals: if \(\tau_\rho<1/2\), then \(k^*=m\) and \(r^*=1\); if \(1/2\le \tau_\rho<2/3\), then \[ k^*=m-\nu(G),\qquad r^*=1-\frac{\nu(G)}{m}; \] if \(1-1/n\le \tau_\rho<1\), then \[ k^*=m-n+\mathrm{cc}(G),\qquad r^*=\frac{m-n+\mathrm{cc}(G)}{m}; \] and if \(\tau_\rho=1\), then \[ k^*=m-n,\qquad r^*=1-\frac{n}{m}. \] 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 [2404.08599]. A complete complexity dichotomy is then given for \(\tau\)-Bounded-Density Edge Deletion: if \(2\tau\in\mathbb N\) or \(\tau<2/3\), the problem is polynomial-time solvable; otherwise it is NP-hard. For integral \(\tau\in\mathbb N\), the paper gives a randomized \(O(m^{1+o(1)})\)-time algorithm, and for fixed rational \(\tau=a/b\), it proves fixed-parameter tractability with respect to treewidth in time \[ (\max\{a,b\}+1)^{\mathrm{tw}(G)+1}\cdot n^{O(1)} \] [2601.03129]. In epidemic control, the same ratio becomes an intervention-intensity variable. For the SIS edge-deletion algorithm, with candidate set \(Q\subseteq E\) and removed set \(P\subseteq Q\), the paper defines \[ r:=\frac{|E_{\text{removed}}|}{|E|}=\frac{|P|}{|E|}. \] The post-deletion linear upper-bounding matrix is \[ M_{-P}:=I-hD+hB_{-P}, \] and if \(\rho(M_{-P})<1\), then \(x(t)\to 0\). In the homogeneous case \(B=\beta A\), \(D=\delta I\), \[ \rho(M)=1-h\delta+h\beta\,\lambda_{\max}(A),\qquad \rho(M)<1 \iff \beta\,\lambda_{\max}(A)<\delta \iff \frac{\beta}{\delta}<\frac{1}{\lambda_{\max}(A)}. \] The greedy algorithm selects edges to maximize the marginal reduction of the supermodular upper bound \[ \hat{\sigma}(P):=\mathbf{1}^{\top}\,(M_{-P} + hD - I)\,(I - M_{-P})^{-1}\,x(0). \] In the reported experiments, the Erdős–Rényi network used \(k=500\) deletions out of \(|E|=5{,}568\), giving \[ r_{\mathrm{ER}}=\frac{500}{5{,}568}\approx 0.0898, \] while the Haslemere dataset used \(k=210\) deletions out of \(|E|=1{,}162\), giving \[ r_{\mathrm{Haslemere}}=\frac{210}{1{,}162}\approx 0.1807. \] Both configurations converged to the disease-free equilibrium under the chosen parameters, and increasing \(r\) reduced cumulative infections with diminishing returns [2606.30142]. ## 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 \(k\). The paper defines the “Edge-Deletion Ratio” as the approximation factor \(\alpha(k+1)\), where \(\alpha\) is determined by a randomized rounding analysis on an LP relaxation. The main bounds are: \[ (2-\sqrt{2})(k+1)\approx 0.585(k+1) \] for i.i.d. uniform labels, \[ 0.549(k+1) \] for a refined independent label distribution, a barrier showing that no independent distribution can improve the analysis below \[ 0.542(k+1), \] and a \[ 0.5(k+1) \] approximation for bipartite DAGs and for instances with structured LP solutions [2507.07943]. 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 \(Y\to X\) and replacing the CPT of \(X\) by \[ \Theta'_{X|U}=\sum_y \Theta_{X|YU}\Pr(y|e). \] A practical definition aligned with the method is the global ratio \[ r=\frac{|E_{\mathrm{del}}|}{|E|}, \] the per-node incoming deletion ratio \[ r_X=\frac{|\{\text{deleted parents of }X\}|}{\operatorname{indegree}(X)}, \] and the evidence-weighted ratio \[ r_{\mathrm{EW}}=\frac{\sum_{Y\to X\in E_{\mathrm{del}}}\mathrm{ENT}(Y|e)} {\sum_{Y\to X\in E}\mathrm{ENT}(Y|e)}. \] The key KL-divergence bound for a single deleted edge is \[ \mathrm{KL}\bigl(\Pr(\cdot|e),\Pr'(\cdot|e)\bigr) \le \log \frac{\Pr'(e)}{\Pr(e)}+\mathrm{ENT}(Y|e), \] and for multiple deleted edges, under the stated restriction of at most one incoming deleted edge per node, \[ \mathrm{KL}\bigl(\Pr(\cdot|e),\Pr'(\cdot|e)\bigr) \le \log \frac{\Pr'(e)}{\Pr(e)}+\sum_{Y\to X}\mathrm{ENT}(Y|e). \] Appendix B shows that the evidence term can be unbounded even when \(\mathrm{ENT}(Y|e)\approx 0\), so a small entropy-based ratio does not by itself guarantee a small approximation error [1207.1370]. ## 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 \(\varepsilon(G)\), the paper defines \[ R_\varepsilon(G,e):=\frac{E_\varepsilon(G-e)}{E_\varepsilon(G)},\qquad \Delta_\varepsilon(G,e):=E_\varepsilon(G-e)-E_\varepsilon(G). \] For the complete \(k\)-partite graph \(K_{n_1,\dots,n_k}\) with \(k\ge 2\) and \(n_i\ge 2\), the paper proves that for any edge \(e\), \[ \Delta_\varepsilon\big(K_{n_1,\dots,n_k},e\big)>0, \] hence \(R_\varepsilon>1\). In the complete bipartite case, examples include \[ R_\varepsilon(K_{2,2},e)=\frac{10}{8}=1.25,\qquad R_\varepsilon(K_{2,3},e)\approx 1.1541,\qquad R_\varepsilon(K_{4,4},e)\approx 1.0354. \] The same paper also gives examples where eccentricity energy and distance energy move in opposite directions under the same deletion [2107.03237]. For degree-based weighted adjacency matrices \(A_\phi(G)\), the edge-deletion ratio can likewise be the post-/pre-deletion quotient of weighted energy or spectral radius. For complete graphs \(K_n\), with \[ s:=\frac{\phi(d_1,d_n)}{\phi(d_n,d_n)}=\frac{1}{r}, \] the exact formulas are \[ R_E(K_n)= \frac{(n-3)+\sqrt{(n-3)^2+8(n-2)s^2}}{2n-2}, \qquad R_\rho(K_n)= \frac{(n-3)+\sqrt{(n-3)^2+8(n-2)s^2}}{2(n-1)}. \] The paper proves the sufficient condition that, for \(G=K_n\) and \(e\in E(K_n)\), \[ \rho_\phi(K_n\setminus e)<\rho_\phi(K_n) \quad\text{and}\quad E_\phi(K_n\setminus e)