--- title: Isolation Number in Graph Theory url: https://www.emergentmind.com/topics/isolation-number type: topic --- # Isolation Number in Graph Theory The **isolation number** of a graph \(G\), denoted by \(\iota(G)\), is the minimum cardinality of a set \(A\subset V(G)\) such that the subgraph induced by the vertices that are not in the union of the closed neighborhoods of vertices in \(A\) has no edges; equivalently, \(V(G)\setminus N[A]\) is an independent set. In the literature, this invariant is also known as the **vertex-edge domination number**. More generally, for a graph \(F\) or a family \(\mathcal F\), the corresponding \(F\)- or \(\mathcal F\)-isolation number asks for a smallest vertex set whose closed neighborhood intersects every copy of \(F\) or every \(\mathcal F\)-graph in the host graph. This places isolation theory within a broad partial-domination framework that contains ordinary domination as the special case \(F=K_1\) and extends naturally to cycles, paths, cliques, stars, chromatic classes, regular graphs, hypergraphs, graph products, and game variants [1504.08055], [2407.18126], [2601.00104]. ## 1. Definition and conceptual framework For a graph \(G=(V,E)\) and a set \(D\subseteq V\), the closed neighborhood is \[ N[D]=D\cup N(D), \] and the remainder is \[ R(D)=V(G)\setminus N[D]. \] A set \(S\subseteq V\) is called \(F\)-isolating for a family of graphs \(F\) if the induced subgraph on the non-dominated vertices, \[ G[R(S)] = G[V\setminus N[S]], \] contains no member of \(F\) as a subgraph. The minimum size of such a set is denoted \(\iota(G,F)\) and is called the \(F\)-isolation number [1504.08055]. When \(F=\{K_2\}\), one obtains the usual isolation number \(\iota(G)\): an isolating set is a set \(D\subseteq V(G)\) such that \(G-N[D]\) has no edges, or equivalently \(V(G)\setminus N[D]\) is independent [1504.08055], [2501.02879], [2508.21551]. When \(F=\{K_1\}\), isolation coincides with domination, \[ \iota(G,K_1)=\gamma(G), \] because \(G-N[D]\) contains no \(K_1\) exactly when it has no vertices [2407.18126], [2303.13709], [2602.22856]. When \(F=\{K_{1,k+1}\}\), the remainder induces a graph of maximum degree at most \(k\), giving the \(k\)-independent or \(k\)-isolation viewpoint [1504.08055], [2408.14653]. This framework is explicitly presented as a form of **constrained partial domination**: instead of requiring that all vertices be dominated, one requires only that the undominated remainder avoid a prescribed configuration. The resulting theory interpolates between domination and a wide range of structural deletion problems, including edge elimination, cycle elimination, bounded-degree remainders, and \(K_k\)-free or \(k\)-colorable remainders [1504.08055], [2303.13709]. ## 2. Fundamental bounds and extremal connected graphs A foundational connected-graph bound states that if \(G\) is a connected graph on \(n\ge 3\) vertices and \(G\neq C_5\), then \[ \iota(G)\le \frac{n}{3}, \] and this bound is sharp [1504.08055]. The same result is stated in equivalent form as: for a connected graph \(G\) on \(n\ge 6\) vertices, there exists a set \(D\subseteq V(G)\) with \[ |D|\le \frac{n}{3} \] such that \[ V(G)\setminus N[D] \] is an independent set [1504.08055]. Subsequent work determined families attaining the extremal ratio \(n/3\). A general family \(\mathcal G\) of connected graphs satisfies \[ \iota(G)=\frac{n}{3}, \] and for specific classes this extremality has been characterized completely. For trees, the equality case is the family \(\mathcal T\); for block graphs, it is the family \(\mathcal B\); and for unicyclic graphs the extremal graphs are \(\mathcal U\cup\{C_6,C_9\}\) [2307.11520]. These characterizations are structurally analogous to classical domination extremal families, but the local gadgets are \(P_3\), \(C_3\), and several six-vertex configurations rather than copies of \(K_2\) [2307.11520]. Leaf-sensitive refinements sharpen the order bound in trees and connected graphs. If \(n\) and \(\ell\) are the number of vertices and the number of leaves of a connected graph \(G\), then for every \(k\ge 1\), \[ \iota_k(G)\le \frac{n-\ell}{2}, \] and if \(T\) is a tree, then \[ \iota(T)\le \frac{n+\ell}{4},\qquad \iota_k(T)\le \frac{n+\ell}{2k+1}\ \text{ for }k\ge 2. \] Each of these bounds is attainable if it is an integer, and the extremal graphs are characterized [2408.14653]. Minimum-degree restrictions yield a different line of refinement. If \(G\) has order \(n\) and minimum degree at least \(4\), then \[ \iota(G)\le \frac{13}{41}n, \] and if \(G\) is also triangle-free, then \[ \iota(G)\le \frac{3}{10}n. \] These bounds are obtained through an isolation residual graph and a weighted potential argument [2508.21551]. ## 3. Isolation with respect to prescribed forbidden subgraphs The general \(F\)-isolation number \(\iota(G,F)\) is defined by requiring \(G-N[D]\) to contain no copy of \(F\). Equivalently, \(N[D]\) must intersect the vertex set of every \(F\)-copy in \(G\) [2407.18126], [2506.09290]. This unifies a substantial part of recent isolation theory. For connected \(m\)-edge graphs \(G\) and a connected \(k\)-edge graph \(F\) with \(\gamma(F)=1\), one has \[ \iota(G,F)\le \left\lfloor \frac{m+1}{k+2}\right\rfloor \] unless \(G\) is an \(F\)-copy or \(F\cong P_3\) and \(G\cong C_6\) [2407.18126]. The equality graphs were later determined: if \(k\ge 3\) and \(F\) is a \(k\)-edge graph with \(\gamma(F)=1\), then a connected \(m\)-edge graph \(G\) attains the bound if and only if \(G\) is either a **pure \((m,F)\)-special graph** or a copy of \(F+e\) for some \(e\in E(\overline F)\) [2506.09290]. Several important special cases fit into this scheme. For \(k\)-clique isolation in graphs, the sharp order bound is \[ \iota(G,K_k)\le \frac{n}{k+1} \] for connected \(n\)-vertex graphs \(G\), except for \(K_k\) and, when \(k=2\), \(C_5\); the corresponding edge version is \[ \iota(G,K_k)\le \left\lfloor \frac{m+1}{\binom{k}{2}+2}\right\rfloor \] for connected \(m\)-edge graphs that are not \(k\)-cliques [2601.00104], [2304.10659]. More generally, if \(\mathcal F=\mathcal F_{0,k}\cup\mathcal F_{1,k}\), where \(\mathcal F_{0,k}=\{K_{1,k}\}\) and \(\mathcal F_{1,k}\) is the set of regular graphs of degree at least \(k-1\), then for connected \(n\)-vertex graphs, \[ \iota(G,\mathcal F)\le \frac{n}{k+1} \] unless \(G\) is a \(k\)-clique or \(k=2\) and \(G\) is a \(5\)-cycle; by Brooks’ Theorem, the same inequality holds for \(\mathcal F_{3,k}=\mathcal F_{0,k}\cup\mathcal F_{1,k}\cup\mathcal F_{2,k}\), where \(\mathcal F_{2,k}\) is the set of graphs with chromatic number at least \(k\) [2303.13709]. These results show that isolation theory has both order-based and edge-based branches, and that the extremal constructions are frequently recursive “special graph” families built from copies of the forbidden pattern with designated connection vertices [2304.10659], [2407.18126], [2506.09290]. ## 4. Cycle, path, and clique isolation Cycle isolation has developed into a particularly detailed subtheory. For \(C_3\), earlier results gave \(\iota(G,C_3)\le n/4\) for connected \(G\not\cong C_3\), and also the size bound \[ \iota(G,C_3)\le \frac{m+1}{5} \] for connected \(m\)-edge graphs \(G\not\cong C_3\) [2310.17337]. For \(C_4\), if \(G\not\cong C_4\) is a connected graph of size \(m\), then \[ \iota(G,C_4)\le \frac{m+1}{6}, \] with equality if and only if \[ G\in \{K_4^-\}\cup \mathcal G_4, \] where \(K_4^-\) is the diamond graph [2310.17337]. For the family \(\mathcal C'\) of non-triangle cycles, that is, cycles of length at least \(4\), a connected \(m\)-edge graph \(G\) that is not a \(4\)-cycle satisfies \[ \iota(G,\mathcal C')\le \frac{m+1}{6}. \] Equality holds exactly for pure \((m,C_4)\)-special graphs and \(\{C_4',C_5\}\)-graphs, where \(C_4'\) is the diamond graph [2510.08361]. Because \(\{C_4\}\subseteq \mathcal C'\), this yields the same \((m+1)/6\) bound for \(C_4\)-isolation as a corollary [2510.08361]. Path isolation exhibits a different extremal pattern. For the \(3\)-vertex path \(P_3\), the \(P_3\)-isolation number \(\iota(G,P_3)\) is the size of a smallest set \(D\) such that \(G-N[D]\) contains no \(P_3\); equivalently, \(G-N[D]\) is a matching plus isolated vertices [2506.19149]. If \(f(n)\) denotes the maximum value of \(\iota(G,P_3)\) over all connected \(n\)-vertex graphs having no induced \(6\)-cycles, then \[ f(n)=\left\lfloor \frac{n+1}{4}\right\rfloor, \] so \[ \limsup_{n\to\infty}\frac{f(n)}{n}=\frac14. \] Moreover, if \[ \iota(G,P_3)=\frac{n+1}{4}, \] then for every vertex \(v\in V(G)\), \[ \iota(G-v,P_3)\le \iota(G,P_3)-1, \] a rigidity property that drives the equality analysis [2506.19149]. Clique isolation also extends to uniform hypergraphs. For a connected \(n\)-vertex \(r\)-uniform hypergraph \(H\), with \(2\le r\le k\), the \(K_k^r\)-isolation number satisfies \[ \iota(H,K_k^r)\le \frac{n}{k+1}, \] unless \(H\simeq K_k^r\), or \(k=r=2\) and \(H\simeq C_5\) [2601.00104]. The proof passes to the \(2\)-shadow \(H^{(2)}\): a \(K_k\)-isolating set in the shadow graph is also a \(K_k^r\)-isolating set in the original \(r\)-graph [2601.00104]. ## 5. Structural methods, extremal constructions, and exact models A recurring feature of isolation theory is the existence of explicit extremal constructions. In edge-based theorems for \(F\)-isolation, one writes \[ m+1=q(k+2)+r,\qquad 0\le r