Grundy Domination in Graph Theory

• Grundy domination is defined as the maximum length of a legal closed neighborhood sequence that dominates a finite simple graph.
• It extends classical domination by focusing on dynamic footprinting processes and establishing dualities with zero forcing and linear-algebraic invariants.
• The concept applies to various graph classes, offering exact formulas for regular, planar, and forest graphs while highlighting NP-completeness and algorithmic solutions.

Searching arXiv for recent and foundational papers on Grundy domination and closely related variants. Searching arXiv for work on regular graphs, forests, graph classes, and variant parameters linked to Grundy domination. Grundy domination is the study of longest legal, stepwise domination processes in graphs. For a finite simple graph $G$, a sequence of distinct vertices $S=(v_1,\dots,v_k)$ is legal when each $v_i$ contributes at least one vertex in $N[v_i]$ that was not contained in any earlier closed neighborhood, and the maximum possible length of such a sequence is the Grundy domination number $\gamma_{\rm gr}(G)$ (Brešar et al., 2020). The parameter is dynamic rather than static: unlike the domination number $\gamma(G)$, which minimizes the size of a dominating set, $\gamma_{\rm gr}(G)$ maximizes the length of a legal construction. It is distinct from the Grundy chromatic number, which belongs to greedy coloring rather than domination theory (Khaleghi et al., 2022).

1. Definitions and sequence formalism

Let $G$ be a finite simple graph with vertex set $V(G)$. For $v\in V(G)$, the open neighborhood is $S=(v_1,\dots,v_k)$0 and the closed neighborhood is $S=(v_1,\dots,v_k)$1. A sequence of distinct vertices

$S=(v_1,\dots,v_k)$2

is a closed neighborhood sequence, or legal sequence, if

$S=(v_1,\dots,v_k)$3

A legal sequence is a dominating sequence when its vertex set dominates $S=(v_1,\dots,v_k)$4, and the Grundy domination number is

$S=(v_1,\dots,v_k)$5

Equivalently, $S=(v_1,\dots,v_k)$6 is the maximum length of a dominating sequence (Brešar et al., 2020).

The legality condition is commonly described through footprinting. At step $S=(v_1,\dots,v_k)$7, the set

$S=(v_1,\dots,v_k)$8

consists of the vertices footprinted by $S=(v_1,\dots,v_k)$9; for each such vertex $v_i$0, $v_i$1 is the footprinter of $v_i$2. In a dominating legal sequence every vertex has a unique footprinter. This language is central in counting arguments, in decomposition results, and in the comparison with zero forcing (Herrman et al., 2022).

Grundy domination sits naturally above several classical domination parameters. The inequalities

$v_i$3

appear repeatedly in the literature, where $v_i$4 is the independence number and $v_i$5 the upper domination number. More generally,

$v_i$6

links Grundy domination to irredundance and upper domination (Bacsó et al., 2022).

2. Variants, forcing dualities, and linear-algebraic bounds

Several closely related sequence parameters refine which neighborhoods are used at each step. The most important variants are the Z-, total-, and L-Grundy parameters, together with their zero-forcing duals (Lin, 2017).

Parameter	Legality condition for $v_i$7	Dual identity
$v_i$8	$v_i$9	$N[v_i]$0
$N[v_i]$1	$N[v_i]$2	$N[v_i]$3
$N[v_i]$4	$N[v_i]$5	$N[v_i]$6
$N[v_i]$7	$N[v_i]$8	$N[v_i]$9

A particularly important variant is the Z-Grundy domination number. If $\gamma_{\rm gr}(G)$0 has no isolated vertices, a sequence $\gamma_{\rm gr}(G)$1 is a Z-sequence if each $\gamma_{\rm gr}(G)$2 footprints a new neighbor distinct from itself, equivalently

$\gamma_{\rm gr}(G)$3

Then

$\gamma_{\rm gr}(G)$4

so maximizing $\gamma_{\rm gr}(G)$5 is equivalent to minimizing the zero forcing number $\gamma_{\rm gr}(G)$6 (Brešar et al., 2020).

For Grundy domination itself, loop zero forcing plays the analogous role. For connected simple graphs,

$\gamma_{\rm gr}(G)$7

where $\gamma_{\rm gr}(G)$8 is the loop zero forcing number. This duality is the basis of several exact results for planar families and claw-free cubic graphs (Domat et al., 2022).

These dualities connect Grundy domination to minimum-rank theory. The inequalities

$\gamma_{\rm gr}(G)$9

bound Grundy-type parameters by minimum-rank-type invariants. In particular, $\gamma(G)$0 is controlled by linear-algebraic data from $\gamma(G)$1, the family of symmetric matrices whose off-diagonal pattern is described by $\gamma(G)$2 and whose diagonal entries are nonzero (Lin, 2017).

3. Regular, cubic, and planar graph bounds

A central regular-graph result is the lower bound

$\gamma(G)$3

for every connected $\gamma(G)$4-regular graph $\gamma(G)$5 of order $\gamma(G)$6, with $\gamma(G)$7, different from $\gamma(G)$8 and $\gamma(G)$9. For $\gamma_{\rm gr}(G)$0, this becomes

$\gamma_{\rm gr}(G)$1

and the connected cubic graphs attaining equality are exactly

$\gamma_{\rm gr}(G)$2

The same work establishes a sharp cubic zero-forcing threshold: if $\gamma_{\rm gr}(G)$3 is a connected cubic graph of order $\gamma_{\rm gr}(G)$4 different from $\gamma_{\rm gr}(G)$5 and $\gamma_{\rm gr}(G)$6, then $\gamma_{\rm gr}(G)$7, with equality characterized by a structured family $\gamma_{\rm gr}(G)$8 together with

$\gamma_{\rm gr}(G)$9

These results are obtained by constructing long legal sequences and controlling the number of new vertices footprinted at each step (Brešar et al., 2020).

Loop-zero-forcing duality yields exact Grundy domination values in several planar classes. For serpentine maximal outerplanar graphs,

$G$0

hence

$G$1

If $G$2 is a maximal outerplanar graph with $G$3, then $G$4 when $G$5 and $G$6 when $G$7, so

$G$8

respectively. For a Halin graph $G$9,

$V(G)$0

and if $V(G)$1 has no vertex $V(G)$2 such that $V(G)$3, then

$V(G)$4

These formulas derive from the exact identity $V(G)$5 (Domat et al., 2022).

4. Forests, products, and graph operations

Forests admit a precise structural formula. If $V(G)$6 is a forest and $V(G)$7 is a minimum caterpillar partition of the non-isolated vertices of $V(G)$8, then

$V(G)$9

Here each part of $v\in V(G)$0 induces a caterpillar, and branch edges between parts must be incident to a non-leaf in at least one of the two caterpillars. In particular, for a path $v\in V(G)$1,

$v\in V(G)$2

The same work proves the strong product formula

$v\in V(G)$3

for any forest $v\in V(G)$4 and any graph $v\in V(G)$5, verifies the strong product conjecture for all forests, and shows that every connected graph $v\in V(G)$6 has a spanning tree $v\in V(G)$7 with

$v\in V(G)$8

It also proves that every connected non-complete graph contains a Grundy dominating set $v\in V(G)$9 such that the induced subgraph $S=(v_1,\dots,v_k)$00 has no isolated vertices (Bell et al., 2021).

Product-graph and substitution frameworks extend this picture. For the $S=(v_1,\dots,v_k)$01-join product $S=(v_1,\dots,v_k)$02, if $S=(v_1,\dots,v_k)$03 is known for each fiber and the main factor $S=(v_1,\dots,v_k)$04 is a power of a cycle, a power of a path, or a split graph, then $S=(v_1,\dots,v_k)$05 can be computed in polynomial time. As a consequence, one obtains closed formulas for the lexicographic product $S=(v_1,\dots,v_k)$06 when $S=(v_1,\dots,v_k)$07 is a power of a cycle, a power of a path, or a split graph (Nasini et al., 2018).

Exact formulas are also known for highly structured graph classes. For the Sierpiński graph $S=(v_1,\dots,v_k)$08,

$S=(v_1,\dots,v_k)$09

For the cycle $S=(v_1,\dots,v_k)$10,

$S=(v_1,\dots,v_k)$11

These formulas come with explicit constructions of Grundy dominating sequences (Bresar et al., 2016).

The parameter is stable under small graph modifications in a sharply quantified way. For every edge $S=(v_1,\dots,v_k)$12,

$S=(v_1,\dots,v_k)$13

and for every vertex $S=(v_1,\dots,v_k)$14,

$S=(v_1,\dots,v_k)$15

If $S=(v_1,\dots,v_k)$16 is simplicial, then

$S=(v_1,\dots,v_k)$17

and if $S=(v_1,\dots,v_k)$18 is a twin vertex, then

$S=(v_1,\dots,v_k)$19

These inequalities are sharp (Bresar et al., 2016).

5. Complexity, exact computation, and equality classes

The decision version of Grundy domination is NP-complete in general, and this remains true for several restrictive graph classes. It is NP-complete even for chordal graphs, and later work proved NP-completeness for bipartite graphs and for co-bipartite graphs. On the positive side, there is a linear-time algorithm for chain graphs, a subclass of bipartite graphs; in this class,

$S=(v_1,\dots,v_k)$20

The same paper observes that co-chain graphs satisfy

$S=(v_1,\dots,v_k)$21

where $S=(v_1,\dots,v_k)$22 is the number of closed-twin layers in the co-chain decomposition (Brešar et al., 2023).

Interval graphs admit a direct exact algorithm. Given an interval representation, one can compute a Grundy dominating sequence in linear time by scanning the interval-endpoint sequence; with preprocessing, the total running time is

$S=(v_1,\dots,v_k)$23

and the scan itself is $S=(v_1,\dots,v_k)$24. Moreover, $S=(v_1,\dots,v_k)$25 equals the number of consecutive subsequences of the form $S=(v_1,\dots,v_k)$26 in the sorted interval-endpoint sequence (Bresar et al., 2016).

A substantial body of work studies when Grundy domination coincides with more classical invariants. Let $S=(v_1,\dots,v_k)$27 denote the class of twin-free connected graphs with

$S=(v_1,\dots,v_k)$28

and $S=(v_1,\dots,v_k)$29 the subclass with

$S=(v_1,\dots,v_k)$30

For every positive integer $S=(v_1,\dots,v_k)$31,

$S=(v_1,\dots,v_k)$32

so hypercubes lie in $S=(v_1,\dots,v_k)$33. Complete multipartite graphs $S=(v_1,\dots,v_k)$34 with $S=(v_1,\dots,v_k)$35 and $S=(v_1,\dots,v_k)$36 satisfy

$S=(v_1,\dots,v_k)$37

whereas prisms $S=(v_1,\dots,v_k)$38 satisfy

$S=(v_1,\dots,v_k)$39

For triangle-free connected graphs in $S=(v_1,\dots,v_k)$40, the graph is bipartite and either $S=(v_1,\dots,v_k)$41 or the graph has a unique $S=(v_1,\dots,v_k)$42-set. The paper culminates in Property U, a characterization of connected graphs $S=(v_1,\dots,v_k)$43 with $S=(v_1,\dots,v_k)$44 (Bacsó et al., 2022).

6. $S=(v_1,\dots,v_k)$45-Grundy domination and open directions

A recent extension replaces the requirement of “never dominated before” by “dominated fewer than $S=(v_1,\dots,v_k)$46 times.” A sequence $S=(v_1,\dots,v_k)$47 is a $S=(v_1,\dots,v_k)$48-sequence if, for each $S=(v_1,\dots,v_k)$49, there exists $S=(v_1,\dots,v_k)$50 such that $S=(v_1,\dots,v_k)$51 appears in the closed neighborhoods of fewer than $S=(v_1,\dots,v_k)$52 previous vertices. The maximum length is the $S=(v_1,\dots,v_k)$53-Grundy domination number

$S=(v_1,\dots,v_k)$54

and $S=(v_1,\dots,v_k)$55. Analogous $S=(v_1,\dots,v_k)$56-$S=(v_1,\dots,v_k)$57, $S=(v_1,\dots,v_k)$58-$S=(v_1,\dots,v_k)$59, and $S=(v_1,\dots,v_k)$60-$S=(v_1,\dots,v_k)$61 variants are defined in parallel, and all are monotone in $S=(v_1,\dots,v_k)$62: $S=(v_1,\dots,v_k)$63 The general comparison inequalities

$S=(v_1,\dots,v_k)$64

extend the familiar $S=(v_1,\dots,v_k)$65 hierarchy (Herrman et al., 2022).

Exact formulas are available for several families. For complete graphs $S=(v_1,\dots,v_k)$66 and $S=(v_1,\dots,v_k)$67,

$S=(v_1,\dots,v_k)$68

For complete bipartite graphs $S=(v_1,\dots,v_k)$69 with $S=(v_1,\dots,v_k)$70,

$S=(v_1,\dots,v_k)$71

$S=(v_1,\dots,v_k)$72

For grids,

$S=(v_1,\dots,v_k)$73

For hypercubes,

$S=(v_1,\dots,v_k)$74

with

$S=(v_1,\dots,v_k)$75

The degree-based bound

$S=(v_1,\dots,v_k)$76

implies

$S=(v_1,\dots,v_k)$77

The paper also proves

$S=(v_1,\dots,v_k)$78

where $S=(v_1,\dots,v_k)$79 is the $S=(v_1,\dots,v_k)$80-forcing number, and conjectures equality (Herrman et al., 2022).

Several open problems remain central. For regular graphs, the unresolved problem is to determine all $S=(v_1,\dots,v_k)$81-regular graphs $S=(v_1,\dots,v_k)$82 of order $S=(v_1,\dots,v_k)$83 with

$S=(v_1,\dots,v_k)$84

for $S=(v_1,\dots,v_k)$85; the cubic case is known, but higher degrees are not (Brešar et al., 2020). For $S=(v_1,\dots,v_k)$86-Grundy domination, open directions include characterizing graphs with $S=(v_1,\dots,v_k)$87, sharpening formulas for hypercubes, and establishing whether

$S=(v_1,\dots,v_k)$88

holds for all graphs (Herrman et al., 2022). These problems indicate that Grundy domination has developed into a broad framework linking greedy domination processes, forcing dynamics, graph products, and minimum-rank methods.