Game Isolation Number in Graphs

• Game Isolation Number is a graph parameter defined via a vertex-selection game where Dominator and Staller alternate moves to isolate an edgeless residual graph.
• It extends classic domination concepts by employing marked graphs and the Continuation Principle to derive bounds and compare with static isolation numbers.
• Exact values on paths and cycles reveal a 2n/5 asymptotic behavior and modular periodicity, while comparisons to edge and total isolation games highlight its nuanced dynamics.

Searching arXiv for the primary papers on the graph-theoretic game isolation number and closely related variants. Search query: "(Brešar et al., 2024) Isolation game on graphs" The game isolation number most commonly denotes the Dominator-start value of the isolation game on a graph $G$, written $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$. In this game, two players, Dominator and Staller, alternately choose vertices, and a move is legal precisely when it dominates a vertex belonging to a nontrivial component of the residual graph $G\setminus N[X]$, where $X$ is the set of already played vertices; Dominator minimizes the number of played vertices and Staller maximizes it. The resulting parameter is a game-theoretic analogue of the static isolation number $\iota(G)$, but the phrase has also been used for distinct invariants in the edge-claiming Toucher–Isolator game and in the total isolation game, so terminological precision is essential (Brešar et al., 2024).

1. Standard definition and formal game model

In the general $\mathcal F$-isolation framework, let $G=(V,E)$ be a graph and $\mathcal F$ a family of graphs. A set $S\subseteq V(G)$ is $\mathcal F$-isolating if $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$0 is $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$1-forbidden, that is, contains no member of $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$2 as a subgraph. The game version is played by Dominator and Staller, who alternately choose vertices. If $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$3 is the set of already played vertices, then a vertex $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$4 is playable if it dominates some vertex lying in a component of $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$5 that is not $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$6-forbidden; the game ends when no playable vertex exists, and the played set is then an $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$7-isolating set (Brešar et al., 2024).

The isolation game is the special case $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$8. In that case, the residual graph $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$9 must be edgeless at termination, so the legal-move condition simplifies: a vertex is playable exactly when it dominates a vertex in a nontrivial component of $G\setminus N[X]$0. The Dominator-start and Staller-start values are denoted

$G\setminus N[X]$1

This is the standard graph-theoretic meaning of “game isolation number” in current usage (Brešar et al., 2024).

A useful reformulation employs partially marked graphs. In a position $G\setminus N[X]$2, a vertex is marked if it is already dominated or lies in a component of $G\setminus N[X]$3 that is already $G\setminus N[X]$4-forbidden. For the isolation game, a basic observation is that a vertex is not playable if and only if its entire closed neighborhood is marked. This marked-graph perspective underlies the main monotonicity and comparison principles (Brešar et al., 2024).

2. Relation to static isolation and domination

The static precursor is the isolation number

$G\setminus N[X]$5

introduced as a form of partial domination. More generally, for a forbidden family $G\setminus N[X]$6,

$G\setminus N[X]$7

Thus the game parameter extends a pre-existing optimization invariant by turning the construction of an isolating set into an adversarial process (Caro et al., 2015).

This framework interpolates naturally with domination. When $G\setminus N[X]$8, the condition $G\setminus N[X]$9 contains no $X$0 means that $X$1 is a dominating set, and the game reduces exactly to the domination game: $X$2 When $X$3, one recovers the isolation game. This places the game isolation number inside the same general methodology as game domination, while preserving a distinct terminal condition: the residual graph must be independent, not empty (Brešar et al., 2024).

The static and game parameters are quantitatively linked. For every graph $X$4 and family $X$5,

$X$6

and

$X$7

For the isolation game, these inequalities show that adversarial play inflates the static optimum by at most a factor of approximately $X$8, while never improving on it (Brešar et al., 2024).

3. Structural principles and universal bounds

A central tool is the Continuation Principle. If $X$9 with $\iota(G)$0, then

$\iota(G)$1

Starting from a more advanced marked position cannot prolong the game. As in domination-game theory, this yields the start-player gap bound

$\iota(G)$2

For $\iota(G)$3, the Dominator-start and Staller-start game isolation numbers therefore differ by at most $\iota(G)$4 (Brešar et al., 2024).

The family-based formulation also gives a comparison theorem. If for every $\iota(G)$5 there exists $\iota(G)$6 such that $\iota(G)$7 is a subgraph of $\iota(G)$8, then

$\iota(G)$9

for every graph $\mathcal F$0. Specializing to $\mathcal F$1 and $\mathcal F$2 yields

$\mathcal F$3

Hence the isolation game never lasts longer than the domination game, although the difference can be arbitrarily large (Brešar et al., 2024).

The principal universal estimate is

$\mathcal F$4

A conjectured sharp replacement is

$\mathcal F$5

This conjecture is supported by infinite sharpness families. One construction, denoted $\mathcal F$6, satisfies

$\mathcal F$7

showing that a $\mathcal F$8 bound, if true, would be best possible on an infinite class (Brešar et al., 2024).

The parameter also exhibits a strong non-monotonicity phenomenon with respect to spanning subgraphs: for any $\mathcal F$9, there exist a graph $G=(V,E)$0 and a spanning subgraph $G=(V,E)$1 such that

$G=(V,E)$2

A plausible implication is that legal-move structure, rather than simple edge density, governs the invariant (Brešar et al., 2024).

4. Exact values for paths and cycles

The path and cycle cases are the sharpest exactly solved families. Earlier work established path bounds differing by at most $G=(V,E)$3 and proved exact values for $G=(V,E)$4 in three residue classes modulo $G=(V,E)$5; later work completed the remaining path classes and determined both start versions on all cycles (Brešar et al., 2024, Bujtás et al., 11 Jul 2025).

The complete formulas are as follows.

Graph class	Dominator-start value	Staller-start value
$G=(V,E)$6	$G=(V,E)$7 if $G=(V,E)$8, and $G=(V,E)$9 otherwise	$\mathcal F$0
$\mathcal F$1	$\mathcal F$2 if $\mathcal F$3, and $\mathcal F$4 otherwise	$\mathcal F$5 if $\mathcal F$6, and $\mathcal F$7 otherwise

These formulas imply

$\mathcal F$8

and similarly for cycles. The modulus-$\mathcal F$9 periodicity reflects the local packing geometry of legal moves on linear and cyclic neighborhoods (Bujtás et al., 11 Jul 2025).

The proofs combine lower-bound strategies for Staller with upper-bound spacing strategies for Dominator. On paths and cycles, the analysis uses runs, meaning maximal sequences of consecutive played vertices. Staller’s prolonging strategy is to play adjacent to an already played vertex whenever possible, which limits the number of runs while ensuring that gaps between runs remain short. Dominator’s complementary strategy is to play at distance $S\subseteq V(G)$0 from an already played vertex, thereby making blocks of four previously playable vertices unplayable. This five-vertex amortization is the structural source of the $S\subseteq V(G)$1 scale (Brešar et al., 2024, Bujtás et al., 11 Jul 2025).

5. Forests, trees, and extremal families

Forests display a start-player asymmetry not present in general graphs. If $S\subseteq V(G)$2 is a partially marked forest, then

$S\subseteq V(G)$3

In particular, for every forest $S\subseteq V(G)$4,

$S\subseteq V(G)$5

This excludes the phenomenon, possible on nonforests, that the Dominator-start game lasts longer than the Staller-start game (Brešar et al., 2024).

For trees, the general $S\subseteq V(G)$6 bound has been improved to

$S\subseteq V(G)$7

The proof introduces the isolation residual graph $S\subseteq V(G)$8, whose vertices are colored white, blue, or red according to whether they still belong to nontrivial components of $S\subseteq V(G)$9, lie in $\mathcal F$0 adjacent to white vertices, or are already irrelevant. On trees, the argument uses a staged weight scheme: initially

$\mathcal F$1

later refined by distinguishing light blue vertices, and then shows that the average weight decrease is at least $\mathcal F$2 per move. Since the initial weight is $\mathcal F$3, this yields the $\mathcal F$4 bound (Bujtás et al., 11 Jul 2025).

The same paper sharpened the universal $\mathcal F$5 theorem by characterizing equality. If $\mathcal F$6 is connected of order $\mathcal F$7, then

$\mathcal F$8

with equality if and only if $\mathcal F$9. For the Staller-start version,

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$00

with equality if and only if $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$01 belongs to a family of precisely eleven connected graphs $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$02 displayed in Figure 1 of the paper (Bujtás et al., 11 Jul 2025).

An additional extremal family supports the conjectural $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$03 scale. For every graph $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$04, let $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$05 be obtained by attaching to each vertex two disjoint triangles and joining the original vertex by one edge to a vertex of each triangle. Then

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$06

This gives an infinite family with exact value at the conjectured sharp constant (Bujtás et al., 11 Jul 2025).

6. Alternative usages and related game parameters

A persistent source of confusion is that “game isolation number” is not universally reserved for $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$07. In the Toucher–Isolator game, the board is $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$08, Toucher and Isolator alternately claim edges, Toucher moves first, and the central parameter is

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$09

A vertex is untouched if none of its incident edges is claimed by Toucher. This is an edge-claiming Maker–Breaker type game rather than the vertex-selection isolation game, and its invariant is therefore not $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$10 (Dowden et al., 2019).

The tree theory of this edge game is especially developed. For a tree $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$11 on $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$12 vertices,

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$13

and this bound is sharp because

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$14

The path is extremal for the minimum of $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$15 over trees of order $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$16, although equality is not unique: the family $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$17, obtained from a path on $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$18 vertices by adding one extra leaf to the second vertex, also satisfies $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$19 (Raty, 2020).

A further variant is the total isolation game, whose parameter is $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$20. Here legal moves are defined using open neighborhoods and total domination constraints rather than closed neighborhoods and ordinary domination. Unlike the ordinary isolation game, the Continuation Principle fails in this setting. For connected graphs of order $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$21,

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$22

while if the minimum degree satisfies $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$23, then

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$24

and if $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$25, then

$\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$26

These results concern a distinct, stronger total analogue and should not be conflated with $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$27 (Henning et al., 6 Jan 2026).

In current graph-game literature, the standard meaning of game isolation number is therefore $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$28: the optimal length of the Dominator-start isolation game on vertices. The static isolation number $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$29, the edge-game parameter $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$30, and the total version $\iota_{\rm g}(G)=\iota_{\rm g}(G,\{K_2\})$31 are all related, but none is interchangeable with it (Brešar et al., 2024).