Isolation Number in Graph Theory

• Isolation number is defined as the minimum size of a vertex set such that the undominated vertices form an independent set.
• It generalizes domination by considering the closed neighborhood covering specific forbidden subgraphs, including cliques, cycles, and paths.
• The theory provides sharp order bounds and structural extremal constructions, linking isolation to both edge-based and vertex-based graph parameters.

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 $\iota(G)$0-graph in the host graph. This places isolation theory within a broad partial-domination framework that contains ordinary domination as the special case $\iota(G)$1 and extends naturally to cycles, paths, cliques, stars, chromatic classes, regular graphs, hypergraphs, graph products, and game variants (Caro et al., 2015, Borg, 2024, Borg, 31 Dec 2025).

1. Definition and conceptual framework

For a graph $\iota(G)$2 and a set $\iota(G)$3, the closed neighborhood is

$\iota(G)$4

and the remainder is

$\iota(G)$5

A set $\iota(G)$6 is called $\iota(G)$7-isolating for a family of graphs $\iota(G)$8 if the induced subgraph on the non-dominated vertices,

$\iota(G)$9

contains no member of $A\subset V(G)$0 as a subgraph. The minimum size of such a set is denoted $A\subset V(G)$1 and is called the $A\subset V(G)$2-isolation number (Caro et al., 2015).

When $A\subset V(G)$3, one obtains the usual isolation number $A\subset V(G)$4: an isolating set is a set $A\subset V(G)$5 such that $A\subset V(G)$6 has no edges, or equivalently $A\subset V(G)$7 is independent (Caro et al., 2015, Zhang et al., 6 Jan 2025, Goddard et al., 29 Aug 2025). When $A\subset V(G)$8, isolation coincides with domination,

$A\subset V(G)$9

because $A$0 contains no $A$1 exactly when it has no vertices (Borg, 2024, Borg, 2023, Borg et al., 26 Feb 2026). When $A$2, the remainder induces a graph of maximum degree at most $A$3, giving the $A$4-independent or $A$5-isolation viewpoint (Caro et al., 2015, Borg et al., 2024).

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 $A$6-free or $A$7-colorable remainders (Caro et al., 2015, Borg, 2023).

2. Fundamental bounds and extremal connected graphs

A foundational connected-graph bound states that if $A$8 is a connected graph on $A$9 vertices and $V(G)\setminus N[A]$0, then

$V(G)\setminus N[A]$1

and this bound is sharp (Caro et al., 2015). The same result is stated in equivalent form as: for a connected graph $V(G)\setminus N[A]$2 on $V(G)\setminus N[A]$3 vertices, there exists a set $V(G)\setminus N[A]$4 with

$V(G)\setminus N[A]$5

such that

$V(G)\setminus N[A]$6

is an independent set (Caro et al., 2015).

Subsequent work determined families attaining the extremal ratio $V(G)\setminus N[A]$7. A general family $V(G)\setminus N[A]$8 of connected graphs satisfies

$V(G)\setminus N[A]$9

and for specific classes this extremality has been characterized completely. For trees, the equality case is the family $F$0; for block graphs, it is the family $F$1; and for unicyclic graphs the extremal graphs are $F$2 (Lemanska et al., 2023). These characterizations are structurally analogous to classical domination extremal families, but the local gadgets are $F$3, $F$4, and several six-vertex configurations rather than copies of $F$5 (Lemanska et al., 2023).

Leaf-sensitive refinements sharpen the order bound in trees and connected graphs. If $F$6 and $F$7 are the number of vertices and the number of leaves of a connected graph $F$8, then for every $F$9,

$\mathcal F$0

and if $\mathcal F$1 is a tree, then

$\mathcal F$2

Each of these bounds is attainable if it is an integer, and the extremal graphs are characterized (Borg et al., 2024).

Minimum-degree restrictions yield a different line of refinement. If $\mathcal F$3 has order $\mathcal F$4 and minimum degree at least $\mathcal F$5, then

$\mathcal F$6

and if $\mathcal F$7 is also triangle-free, then

$\mathcal F$8

These bounds are obtained through an isolation residual graph and a weighted potential argument (Goddard et al., 29 Aug 2025).

3. Isolation with respect to prescribed forbidden subgraphs

The general $\mathcal F$9-isolation number $F$0 is defined by requiring $F$1 to contain no copy of $F$2. Equivalently, $F$3 must intersect the vertex set of every $F$4-copy in $F$5 (Borg, 2024, Borg et al., 10 Jun 2025). This unifies a substantial part of recent isolation theory.

For connected $F$6-edge graphs $F$7 and a connected $F$8-edge graph $F$9 with $\mathcal F$0, one has

$\mathcal F$1

unless $\mathcal F$2 is an $\mathcal F$3-copy or $\mathcal F$4 and $\mathcal F$5 (Borg, 2024). The equality graphs were later determined: if $\mathcal F$6 and $\mathcal F$7 is a $\mathcal F$8-edge graph with $\mathcal F$9, then a connected $F$0-edge graph $F$1 attains the bound if and only if $F$2 is either a pure $F$3-special graph or a copy of $F$4 for some $F$5 (Borg et al., 10 Jun 2025).

Several important special cases fit into this scheme. For $F$6-clique isolation in graphs, the sharp order bound is

$F$7

for connected $F$8-vertex graphs $F$9, except for $\iota(G)$00 and, when $\iota(G)$01, $\iota(G)$02; the corresponding edge version is

$\iota(G)$03

for connected $\iota(G)$04-edge graphs that are not $\iota(G)$05-cliques (Borg, 31 Dec 2025, Borg, 2023). More generally, if $\iota(G)$06, where $\iota(G)$07 and $\iota(G)$08 is the set of regular graphs of degree at least $\iota(G)$09, then for connected $\iota(G)$10-vertex graphs,

$\iota(G)$11

unless $\iota(G)$12 is a $\iota(G)$13-clique or $\iota(G)$14 and $\iota(G)$15 is a $\iota(G)$16-cycle; by Brooks’ Theorem, the same inequality holds for $\iota(G)$17, where $\iota(G)$18 is the set of graphs with chromatic number at least $\iota(G)$19 (Borg, 2023).

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 (Borg, 2023, Borg, 2024, Borg et al., 10 Jun 2025).

4. Cycle, path, and clique isolation

Cycle isolation has developed into a particularly detailed subtheory. For $\iota(G)$20, earlier results gave $\iota(G)$21 for connected $\iota(G)$22, and also the size bound

$\iota(G)$23

for connected $\iota(G)$24-edge graphs $\iota(G)$25 (Wei et al., 2023). For $\iota(G)$26, if $\iota(G)$27 is a connected graph of size $\iota(G)$28, then

$\iota(G)$29

with equality if and only if

$\iota(G)$30

where $\iota(G)$31 is the diamond graph (Wei et al., 2023).

For the family $\iota(G)$32 of non-triangle cycles, that is, cycles of length at least $\iota(G)$33, a connected $\iota(G)$34-edge graph $\iota(G)$35 that is not a $\iota(G)$36-cycle satisfies

$\iota(G)$37

Equality holds exactly for pure $\iota(G)$38-special graphs and $\iota(G)$39-graphs, where $\iota(G)$40 is the diamond graph (Borg et al., 9 Oct 2025). Because $\iota(G)$41, this yields the same $\iota(G)$42 bound for $\iota(G)$43-isolation as a corollary (Borg et al., 9 Oct 2025).

Path isolation exhibits a different extremal pattern. For the $\iota(G)$44-vertex path $\iota(G)$45, the $\iota(G)$46-isolation number $\iota(G)$47 is the size of a smallest set $\iota(G)$48 such that $\iota(G)$49 contains no $\iota(G)$50; equivalently, $\iota(G)$51 is a matching plus isolated vertices (Bartolo et al., 23 Jun 2025). If $\iota(G)$52 denotes the maximum value of $\iota(G)$53 over all connected $\iota(G)$54-vertex graphs having no induced $\iota(G)$55-cycles, then

$\iota(G)$56

so

$\iota(G)$57

Moreover, if

$\iota(G)$58

then for every vertex $\iota(G)$59,

$\iota(G)$60

a rigidity property that drives the equality analysis (Bartolo et al., 23 Jun 2025).

Clique isolation also extends to uniform hypergraphs. For a connected $\iota(G)$61-vertex $\iota(G)$62-uniform hypergraph $\iota(G)$63, with $\iota(G)$64, the $\iota(G)$65-isolation number satisfies

$\iota(G)$66

unless $\iota(G)$67, or $\iota(G)$68 and $\iota(G)$69 (Borg, 31 Dec 2025). The proof passes to the $\iota(G)$70-shadow $\iota(G)$71: a $\iota(G)$72-isolating set in the shadow graph is also a $\iota(G)$73-isolating set in the original $\iota(G)$74-graph (Borg, 31 Dec 2025).

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 $\iota(G)$75-isolation, one writes

$\iota(G)$76

and constructs $\iota(G)$77-special graphs by gluing together $\iota(G)$78 copies of $\iota(G)$79 through designated connection vertices, together with a small remainder graph (Borg, 2024). In the pure case, the remainder has no edges, and the set of all connection vertices is an isolating set of size $\iota(G)$80; the proof shows that every isolating set must meet each $\iota(G)$81-constituent (Borg, 2024, Borg et al., 10 Jun 2025).

Analogous recursive constructions appear for clique isolation, cycle isolation, and hypergraph clique isolation. Pure $\iota(G)$82-special graphs attain the sharp edge-based clique and chromatic/regular isolation bounds (Borg, 2023). Pure $\iota(G)$83-special graphs attain the sharp $\iota(G)$84 bound for non-triangle cycle isolation (Borg et al., 9 Oct 2025). Pure $\iota(G)$85-good $\iota(G)$86-graphs attain

$\iota(G)$87

and for $\iota(G)$88, equality in

$\iota(G)$89

holds if and only if $\iota(G)$90 is either a pure $\iota(G)$91-good $\iota(G)$92-graph or, when $\iota(G)$93, one of the explicit exceptional hypergraphs in $\iota(G)$94 (Borg, 31 Dec 2025).

Structural decomposition lemmas are equally central. Isolation numbers are additive over connected components for connected forbidden families: $\iota(G)$95 when $\iota(G)$96 are the components of $\iota(G)$97 (Wei et al., 2023, Borg, 2024). Another standard reduction states that if $\iota(G)$98, then

$\iota(G)$99

which supports inductive proofs based on deleting a closed neighborhood and analyzing the remaining components (Borg, 2024, Borg et al., 10 Jun 2025).

These methods indicate that the extremal side of isolation theory is not merely an adjunct to upper bounds. In several papers, the determination of equality graphs is explicitly part of the proof architecture, because equality forces every intermediate estimate to be tight and thereby imposes a rigid recursive structure (Borg, 2023, Borg et al., 9 Oct 2025, Borg et al., 10 Jun 2025).

6. Variants, products, complexity, and games

A number of variants modify the admissible isolating set or the ambient graph construction. The independent isolation number requires the isolating set $A\subset V(G)$00 itself to be independent. For general graphs of order $A\subset V(G)$01, this parameter can be arbitrarily close to $A\subset V(G)$02; for connected bipartite graphs on at least three vertices, the vertex set can be partitioned into three independent isolating sets, so the independent isolation number is at most $A\subset V(G)$03; and for connected $A\subset V(G)$04-colorable graphs,

$A\subset V(G)$05

with equality realized by an explicit family $A\subset V(G)$06 (Boyer et al., 12 Mar 2025). This suggests that independence changes the extremal order of magnitude in general graphs but preserves $A\subset V(G)$07-type behavior in structured classes.

Product constructions reveal further links with domination and matching theory. For middle graphs,

$A\subset V(G)$08

where $A\subset V(G)$09 is the size of a smallest maximal matching of $A\subset V(G)$10 (Zhang et al., 6 Jan 2025). For prisms, the inequality

$A\subset V(G)$11

holds, and if $A\subset V(G)$12 is bipartite, then

$A\subset V(G)$13

In particular,

$A\subset V(G)$14

for all positive integers $A\subset V(G)$15 (Bresar et al., 22 Aug 2025). For lexicographic products $A\subset V(G)$16, if $A\subset V(G)$17 and $A\subset V(G)$18 are nontrivial connected graphs and $A\subset V(G)$19, then

$A\subset V(G)$20

(Bresar et al., 22 Aug 2025).

The algorithmic picture is negative in general. If $A\subset V(G)$21 is connected, then the $A\subset V(G)$22-isolating set problem is NP-complete (Borg et al., 26 Feb 2026). The reduction attaches a copy of $A\subset V(G)$23 to each vertex of a graph $A\subset V(G)$24 and proves

$A\subset V(G)$25

thereby transferring hardness from Dominating Set (Borg et al., 26 Feb 2026). The same paper studies minimum-degree asymptotics through

$A\subset V(G)$26

and shows

$A\subset V(G)$27

so the largest possible $A\subset V(G)$28-isolation number in minimum-degree-$A\subset V(G)$29 graphs is asymptotically of order $A\subset V(G)$30 (Borg et al., 26 Feb 2026).

Game versions replace minimization by optimal play. In the isolation game, Dominator and Staller alternately select playable vertices until the played set becomes isolating (Bujtás et al., 11 Jul 2025). For every connected graph $A\subset V(G)$31,

$A\subset V(G)$32

with equality if and only if $A\subset V(G)$33; for trees $A\subset V(G)$34 of order at least $A\subset V(G)$35,

$A\subset V(G)$36

Exact values are known for paths and cycles (Bujtás et al., 11 Jul 2025). A related total isolation game studies total isolating sets and proves, for connected graphs of order $A\subset V(G)$37,

$A\subset V(G)$38

while if $A\subset V(G)$39,

$A\subset V(G)$40

and, more generally,

$A\subset V(G)$41

for diameter-$A\subset V(G)$42 graphs,

$A\subset V(G)$43

(Henning et al., 6 Jan 2026).

Finally, subdivision criticality has recently been introduced. A graph $A\subset V(G)$44 is $A\subset V(G)$45-critical if subdividing any $A\subset V(G)$46 edges increases $A\subset V(G)$47, while some set of $A\subset V(G)$48 edge subdivisions leaves $A\subset V(G)$49 unchanged (Bartolo et al., 26 Feb 2026). For each integer $A\subset V(G)$50 there exists a $A\subset V(G)$51-critical graph, and $A\subset V(G)$52-critical graphs admit a structural characterization via critical tripartitions; for trees, these graphs are described constructively by a recursively defined family $A\subset V(G)$53 (Bartolo et al., 26 Feb 2026).

7. Relationship to domination theory and present scope

Isolation theory is consistently presented as a broad generalization of domination. The identity

$A\subset V(G)$54

is the formal bridge, while the ordinary isolation number $A\subset V(G)$55 is a relaxation in which uncovered vertices may remain provided they induce no edge (Caro et al., 2015, Borg, 2024, Borg et al., 26 Feb 2026). Many upper bounds mirror classical domination bounds: Ore-type order bounds, Arnautov–Lovász–Payan minimum-degree bounds, and tree-structured extremal families all reappear in isolation theory with modified constants and different local obstructions (Borg, 2023, Borg et al., 2024, Borg et al., 26 Feb 2026).

At the same time, the modern theory has moved well beyond the original invariant. There are now sharp results for isolation of cliques, paths, cycles, stars, regular graphs, $A\subset V(G)$56-chromatic graphs, non-triangle cycles, graphs with a universal vertex, and complete uniform hypergraphs; exact formulas for middle graphs, generalized Sierpiński graphs, paths, and cycles in game settings; NP-completeness for every connected forbidden pattern; and refined extremal constructions in both graphs and hypergraphs (Wei et al., 2023, Bartolo et al., 23 Jun 2025, Borg et al., 9 Oct 2025, Bresar et al., 22 Aug 2025, Borg, 31 Dec 2025, Borg et al., 26 Feb 2026). A plausible implication is that isolation number has become a unifying parameter at the intersection of domination, forbidden-subgraph deletion, extremal graph theory, and structural graph algorithms.