Isolating Set: Definitions and Applications

• Isolating sets are defined as selected subsets that, when removed or employed, disrupt connections in systems like computational geometry, graph theory, and dynamics.
• In computational geometry, they minimize unit disk subfamilies to disconnect point sets, with NP-completeness established through reductions from planar multiterminal cut problems.
• In graph theory and dynamics, isolating sets correspond to closed-neighborhood eliminations and isolating neighborhoods that confine invariant behaviors, leading to various structural and game-theoretic variants.

An isolating set is a term whose precise meaning depends on the ambient domain, but its recurring role is to enforce separation by selecting a subset whose presence or removal prevents designated connections, substructures, or trajectories from persisting in the residual object. In computational geometry, a subfamily of unit disks isolates a finite point set when every path between distinct points meets a selected disk; in graph theory, a vertex set isolates a graph or a family of graphs when deleting its closed neighborhood leaves no forbidden subgraph; in terminal-cut problems, the term may refer to the source side of a minimum cut separating one terminal from the others; and in dynamics, the closely related language of isolated invariant sets, isolating neighborhoods, and isolating blocks formalizes local confinement of trajectories (Penninger et al., 2013, Borg et al., 26 Feb 2026, Velednitsky, 2018, Moreira et al., 2023).

1. Semantic range and common abstraction

A common pattern across these usages is residual elimination: one chooses a set so that a complement, a reduced graph, or a reduced state-space region no longer contains the target phenomenon. In graph-theoretic isolation, the residual object is typically $G-N[S]$, where $N[S]$ is a closed neighborhood. In geometric isolation with disks, the relevant residual is $\mathbb{R}^2 \setminus \bigcup D'$. In cut problems, the selected object is itself a separating cut or its source side. In dynamical systems, the ambient set is not removed; rather, one seeks a neighborhood whose invariant part is confined to its interior.

The term is therefore overloaded rather than uniform. In Boolean matrix theory, for example, an isolation set is also called a fooling set: a set of $1$-entries with no two in the same row or column and no two lying in an all-ones $2\times 2$ submatrix (Parnas et al., 2019). In multivalued semiflow theory, by contrast, the central object is an isolated weakly invariant set together with an isolating neighborhood or isolating block, not a hitting set in the combinatorial sense (Moreira et al., 2023).

2. Point-set isolation in computational geometry

For point-set isolation with unit disks, the input consists of a finite set $S \subset \mathbb{R}^2$ of points and a finite family $D$ of congruent closed disks of radius $1$. A disk centered at $c$ is

$B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$

For a subfamily $N[S]$0, writing $N[S]$1, the selected disks isolate $N[S]$2 when every continuous path between any two distinct points of $N[S]$3 intersects $N[S]$4. Equivalently, no two points of $N[S]$5 lie in the same connected component of the complement (Penninger et al., 2013): $N[S]$6

The associated optimization problem minimizes $N[S]$7 under this separation condition, and the decision version asks whether there is such a subfamily with $N[S]$8. The fundamental complexity result is that this problem is NP-complete when $N[S]$9 is not fixed (Penninger et al., 2013). Membership in NP is established by constructing the arrangement induced by the selected disks, performing planar point-location queries for the points of $\mathbb{R}^2 \setminus \bigcup D'$0, and checking whether any pair lies in the same face.

The NP-hardness proof proceeds through a geometric reformulation of planar multiterminal cut. The intermediary Planar Subdivision Problem asks for a minimum edge set whose deletion separates designated points placed in distinct faces of an embedded planar graph. The reduction then replaces each edge by a uniform edge gadget formed from congruent disks in an elongated corridor and each vertex by a ring of disks. Grid-embedding lemmas control non-incidence interactions: the minimum angle between distinct grid lines through a grid point is $\mathbb{R}^2 \setminus \bigcup D'$1, and the minimum distance from a line through two grid points to any other grid point is at least $\mathbb{R}^2 \setminus \bigcup D'$2 (Penninger et al., 2013). One explicit parameter choice satisfying the geometric constraints before scaling is

$\mathbb{R}^2 \setminus \bigcup D'$3

after which a uniform scaling makes all disks unit radius.

The construction is calibrated so that each face of the embedded planar graph corresponds to a unique connected region of $\mathbb{R}^2 \setminus \bigcup D'$4, and deleting a disk from an edge gadget merges precisely the two adjacent regions. By setting the number $\mathbb{R}^2 \setminus \bigcup D'$5 of disks in an edge gadget larger than the total number of disks in all vertex gadgets, optimal solutions are forced to delete disks only from edge gadgets, thereby recovering the planar cut instance (Penninger et al., 2013).

The same paper proves two closely related hardness results. First, the all-cells-connection problem—choosing a minimum $\mathbb{R}^2 \setminus \bigcup D'$6 so that $\mathbb{R}^2 \setminus \bigcup D'$7 is connected—is NP-complete via a reduction from Feedback Vertex Set in planar graphs of maximum degree $\mathbb{R}^2 \setminus \bigcup D'$8. Second, multiterminal cut remains NP-complete on unit disk graphs, even though the graph class is induced by uniform-radius intersection geometry (Penninger et al., 2013).

3. Graph isolation as closed-neighborhood elimination

In graph theory, the standard abstraction is defined relative to a forbidden graph $\mathbb{R}^2 \setminus \bigcup D'$9 or a family $1$0. For a finite simple graph $1$1 and $1$2, the closed neighborhood is

$1$3

A set $1$4 is $1$5-isolating if $1$6 contains no $1$7-graph; equivalently, $1$8 intersects the vertex set of every subgraph isomorphic to a member of $1$9 (Borg et al., 26 Feb 2026). When $2\times 2$0, the minimum size of such a set is denoted $2\times 2$1.

This framework contains domination as the case $2\times 2$2: $2\times 2$3 It also contains the classical isolation number as the case $2\times 2$4, where one requires $2\times 2$5 to be edgeless (Borg et al., 26 Feb 2026).

For every connected graph $2\times 2$6, the decision problem asking whether $2\times 2$7 admits an $2\times 2$8-isolating set of size at most $2\times 2$9 is NP-complete (Borg et al., 26 Feb 2026). The hardness proof is uniform in $S \subset \mathbb{R}^2$0 and reduces Dominating Set to $S \subset \mathbb{R}^2$1-isolation via the $S \subset \mathbb{R}^2$2-corona construction $S \subset \mathbb{R}^2$3, obtained by gluing a copy of $S \subset \mathbb{R}^2$4 to each vertex of $S \subset \mathbb{R}^2$5 at a designated vertex $S \subset \mathbb{R}^2$6. The key identity is

$S \subset \mathbb{R}^2$7

which transfers NP-hardness directly from domination. The same construction preserves several graph classes: for connected $S \subset \mathbb{R}^2$8, $S \subset \mathbb{R}^2$9-isolation is NP-complete on suitable planar, chordal, and bipartite classes (Borg et al., 26 Feb 2026).

The theory also studies extremal behavior as a function of minimum degree. If $D$0 and

$D$1

then domination bounds imply $D$2, where $D$3 is the extremal isolation fraction over graphs of minimum degree at least $D$4. The asymptotically matching lower bound is

$D$5

showing a bounded-range phenomenon controlled by $D$6 and $D$7 (Borg et al., 26 Feb 2026).

For disconnected targets, the paper introduces $D$8, the disjoint union of $D$9 copies of $1$0, and more generally $1$1. The basic inequality is

$1$2

where $1$3 (Borg et al., 26 Feb 2026). If $1$4 has the Erdős–Pósa property with function $1$5, this improves to

$1$6

A notable special case is cycle isolation. Letting $1$7 be the family of cycles and $1$8 the feedback vertex number, subdividing each edge of $1$9 twice gives a graph $c$0 satisfying

$c$1

which ties $c$2-isolation directly to Feedback Vertex Set (Borg et al., 26 Feb 2026).

4. Quantitative, structural, and game-theoretic variants in graphs

For the classical isolation number $c$3, sharper upper bounds are known under degree restrictions. Using an “isolation residual graph” method with white, blue, and red vertices and a weighted potential function, it is shown that if $c$4, then

$c$5

and if $c$6 is triangle-free with $c$7, then

$c$8

The same method yields $c$9 for $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$0, $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$1 for triangle-free graphs with $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$2, and $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$3 when $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$4 and the girth is at least $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$5 (Goddard et al., 29 Aug 2025).

Isolation also admits partition formulations. If $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$6 and $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$7 is connected with $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$8 and $B(c,1)=\{x\in\mathbb{R}^2:\|x-c\|_2\le 1\}.$9, then $N[S]$00 can be partitioned into $N[S]$01 disjoint $N[S]$02-clique isolating sets. If $N[S]$03 is connected, claw-free, subcubic, and $N[S]$04, then $N[S]$05 can be partitioned into four disjoint cycle isolating sets (Zhang et al., 2024). These are isomatic-type statements: they strengthen one-set upper bounds by producing multiple disjoint isolating layers.

Requiring the isolating set itself to be independent leads to the independent isolation number. In general graphs this parameter can be arbitrarily close to $N[S]$06, but in bipartite graphs the vertex set can be partitioned into three disjoint independent isolating sets, implying $N[S]$07. For connected 3-colorable graphs,

$N[S]$08

and for connected $N[S]$09-colorable graphs with $N[S]$10,

$N[S]$11

The existence of three disjoint independent isolating sets is NP-complete (Boyer et al., 12 Mar 2025).

A game version has also been introduced. In the $N[S]$12-isolation game, Dominator and Staller alternately build an $N[S]$13-isolating set, with Dominator minimizing and Staller maximizing the length. The Continuation Principle holds, and the difference between the Dominator-start and Staller-start game values is at most $N[S]$14. For the ordinary isolation game $N[S]$15,

$N[S]$16

and for paths $N[S]$17,

$N[S]$18

(Brešar et al., 2024).

Leaf-sensitive variants refine the dependence on graph structure. For the star-isolation number $N[S]$19, defined so that $N[S]$20 has maximum degree at most $N[S]$21, every connected $N[S]$22-vertex graph with $N[S]$23 leaves satisfies

$N[S]$24

If $N[S]$25 is a tree, then

$N[S]$26

and these bounds are sharp with complete extremal characterizations (Borg et al., 2024). A different tree generalization, all-$N[S]$27-isolation, requires every component of $N[S]$28 to have order strictly less than $N[S]$29. Every tree of order $N[S]$30 has an independent all-$N[S]$31-isolating set of size at most $N[S]$32, equality is characterized by an explicit family $N[S]$33, and for $N[S]$34 there are $N[S]$35 disjoint independent all-$N[S]$36-isolating sets apart from the excluded $N[S]$37 cases and one additional obstruction $N[S]$38 when $N[S]$39 (Boyer et al., 15 Sep 2025).

Target-specific isolation has been developed for cycles and connected graphs with bounded edge counts. For $N[S]$40-cycles, if $N[S]$41 is connected and not isomorphic to one of nine exceptional graphs, then

$N[S]$42

while for connected $N[S]$43 of size $N[S]$44,

$N[S]$45

with equality exactly for $N[S]$46 and the family $N[S]$47 (Bartolo et al., 2023, Wei et al., 2023). For the families $N[S]$48 and $N[S]$49 of connected graphs with at least $N[S]$50 or $N[S]$51 edges, one has

$N[S]$52

and

$N[S]$53

for connected $N[S]$54 not belonging to a six-graph exceptional family, where $N[S]$55 is the number of leaves (Borg, 2021). Extremal equality at the classical $N[S]$56 threshold has also been characterized for unicyclic graphs and block graphs (Lemanska et al., 2023).

5. Isolating cuts and terminal connectivity

In multiway cut theory, an isolating cut for a terminal $N[S]$57 is a minimum cut separating $N[S]$58 from the super-sink formed by the remaining terminals. Its source side is denoted $N[S]$59, and this source side is the corresponding isolating set in the partition sense (Velednitsky, 2018). For the $N[S]$60-Terminal Cut problem, the classical algorithm computes every isolating cut $N[S]$61 and returns the union of all except the largest: $N[S]$62 This yields a $N[S]$63-approximation (Velednitsky, 2018).

Stability sharpens this picture. If the instance is $N[S]$64-stable, then the source sets $N[S]$65 of the isolating cuts coincide with the source sets $N[S]$66 of the unique optimal multiway partition: $N[S]$67 Consequently, the union-of-isolating-cuts algorithm becomes exact on $N[S]$68-stable instances. The threshold is tight: for every $N[S]$69, there exist $N[S]$70-stable instances in which the isolating sets are trivial, $N[S]$71, and the approximation algorithm fails to return the optimum (Velednitsky, 2018).

A more abstract isolating-cut framework is available for symmetric bisubmodular functions. Generalizing the Li–Panigrahi technique, all isolating cuts for a terminal set $N[S]$72 can be computed with $N[S]$73 $N[S]$74-$N[S]$75 minimum-cut computations by intersecting appropriate minimum cuts for a small family of terminal bipartitions (Chekuri et al., 2021). In the corresponding lattice formulation, each terminal $N[S]$76 obtains a region $N[S]$77 containing an $N[S]$78-minimum $N[S]$79-cut, and the first components of the $N[S]$80 are pairwise disjoint. This reduction supports faster randomized algorithms for hypergraph global connectivity, element connectivity, vertex connectivity, and symmetric submodular minimum-cut problems (Chekuri et al., 2021).

These algorithmic results coexist with hardness on restricted geometric graph classes. In particular, multiterminal cut remains NP-complete on unit disk graphs when the number of terminals is not fixed, so geometric intersection structure does not eliminate intrinsic cut complexity (Penninger et al., 2013).

6. Matrix, topological, and non-autonomous dynamical meanings

In the uniform intersection matrix $N[S]$81, whose rows and columns are indexed by $N[S]$82-subsets of $N[S]$83 and whose entries indicate nonempty intersection, an isolation set is a fooling set of $N[S]$84-entries. The largest identity submatrix in $N[S]$85 has size

$N[S]$86

and the paper constructs large isolation sets in two regimes: if $N[S]$87 with $N[S]$88, there exists an isolation set of size $N[S]$89; if $N[S]$90, there exists an isolation set of size $N[S]$91, which is maximal because the Boolean rank of $N[S]$92 is then $N[S]$93 (Parnas et al., 2019). The same work proves that for sufficiently large $N[S]$94, the largest triangular isolation submatrix has size $N[S]$95.

In multivalued semiflow theory, the vocabulary shifts from selecting a hitting set to localizing invariant dynamics. Let $N[S]$96 be a multivalued semiflow on a metric space $N[S]$97, generated by a family $N[S]$98 satisfying axioms (K1)–(K4). For a closed set $N[S]$99, the forward and backward viability sets are

$\mathbb{R}^2 \setminus \bigcup D'$00

$\mathbb{R}^2 \setminus \bigcup D'$01

The invariant part is

$\mathbb{R}^2 \setminus \bigcup D'$02

A closed set $\mathbb{R}^2 \setminus \bigcup D'$03 is an isolated weakly invariant set if it is weakly invariant and maximal among weakly invariant sets in some neighborhood; $\mathbb{R}^2 \setminus \bigcup D'$04 is isolating when $\mathbb{R}^2 \setminus \bigcup D'$05 (Moreira et al., 2023).

An isolating block $\mathbb{R}^2 \setminus \bigcup D'$06 is an isolating neighborhood whose boundary decomposes into ingress points, egress points, and bounce-off points, with exit set $\mathbb{R}^2 \setminus \bigcup D'$07 closed. Under axioms (K1)–(K5) and the existence of a closed $\mathbb{R}^2 \setminus \bigcup D'$08-admissible isolating neighborhood $\mathbb{R}^2 \setminus \bigcup D'$09, every nonempty closed isolated weakly invariant set $\mathbb{R}^2 \setminus \bigcup D'$10 admits an isolating block $\mathbb{R}^2 \setminus \bigcup D'$11 with

$\mathbb{R}^2 \setminus \bigcup D'$12

(Moreira et al., 2023). The construction uses Lyapunov-like functions $\mathbb{R}^2 \setminus \bigcup D'$13 and $\mathbb{R}^2 \setminus \bigcup D'$14 controlling forward and backward escape.

For non-autonomous systems, isolating-neighborhood methods are formulated on the extended space $\mathbb{R}^2 \setminus \bigcup D'$15 or $\mathbb{R}^2 \setminus \bigcup D'$16 via the skew-product flow

$\mathbb{R}^2 \setminus \bigcup D'$17

An isolating neighborhood is then a compact tube $\mathbb{R}^2 \setminus \bigcup D'$18 satisfying $\mathbb{R}^2 \setminus \bigcup D'$19. In the elliptic restricted three-body problem, simplified cylindrical isolating neighborhood boundaries are computed around libration points in the non-uniformly rotating pulsating frame, then used together with a bisection method to compute forward asymptotic trajectories of the isolated invariant set and to track planar and spatial orbits around the libration region (Anderson et al., 2023). The implementation classifies entry and exit by the sign of $\mathbb{R}^2 \setminus \bigcup D'$20 on time-dependent boundaries and validates the boundary behavior by integrating tangent trajectories.

Taken together, these strands show that “isolating set” is not a single invariant but a family of domain-specific separation notions. In combinatorics and optimization, it is usually a selected subset whose closed neighborhood or geometric support destroys residual connections; in terminal connectivity, it is the source side or cutset of a minimum separator; and in dynamics, the analogous language describes neighborhoods and blocks that isolate invariant behavior rather than delete it.