Maximal Independent Domination

• Maximal independent domination is defined as an independent set that is also dominating, ensuring every vertex outside the set is adjacent to at least one set vertex.
• It connects with maximal independent sets and provides sharp bounds for the independent domination number, especially in graphs with bounded degrees.
• Counting these sets involves polynomial methods and optimization formulations, highlighting both NP-hard complexity and efficient solutions in special graph families.

A maximal independent dominating set in a finite simple graph is a subset of vertices that is both independent (no two are adjacent) and dominating (every vertex not in the set is adjacent to some member of the set). This concept is central in extremal combinatorics, algorithmic graph theory, and enumerative combinatorics, owing to its connections to maximal independent sets, domination parameters, structural characterizations, and computational complexity. The independent domination number, and its variants, play key roles in understanding resilience, redundancy, and structural decomposability in networks.

1. Definitions and Basic Notions

A set $W \subseteq V(G)$ is called an independent dominating set if:

• (i) For every $u, v \in W$, $(u,v) \notin E(G)$ (independence).
• (ii) For every $v \in V(G) \setminus W$, there exists $u \in W$ with $(u,v) \in E(G)$ (domination).

Equivalently, independent dominating sets are precisely the maximal independent sets—those independent sets that cannot be extended by adding another vertex while preserving independence.

Let:

• $\gamma_i(G)$ denote the independent domination number, the minimum size of any independent dominating set in $G$.
• $\Gamma_i(G)$ denote the upper independent domination number, the maximum such size.

The generating function for counting independent dominating sets is the independent domination polynomial:

$D_i(G;x) = \sum_{k \geq 0} d_k x^k$

where $d_k$ counts the number of independent dominating sets of size $k$ (Dod, 2016).

2. Structural Results and Extremal Bounds

For any simple graph with maximum degree $\Delta$, there exist sharp upper bounds on the size of minimal independent dominating sets, especially in the case of bounded-degree and extremal graph constructions. Defining $\alpha(\Delta) = \lfloor \Delta^2 / 4 \rfloor$ and $t(\Delta) = \Delta / (\alpha(\Delta) + \Delta)$:

• If $\Delta = 4$ or $\Delta \geq 6$, every connected $n$-vertex graph $G$ with $\Delta(G) \leq \Delta$ satisfies:

$\gamma_i(G) \leq (1 - t(\Delta))(n-1) + 1$

• For other values, tightened case-based bounds apply, e.g., when $\Delta = 5$:

$\gamma_i(G) \leq (5/9) n$

Special extremal graphs, termed $\Delta$-special (built from cliques or cycles with appropriate leaf attachments), attain equality in these bounds (Cho et al., 2022). As $\Delta$ increases, $t(\Delta) \sim 4/\Delta$ becomes small, indicating sparse independent dominating sets are possible in high-degree graphs. These bounds improve the trivial estimate $\gamma_i(G) \leq n - \Delta$ that results from greedy construction.

3. Polynomial and Recurrence Methods

The independent domination polynomial, $D_i(G;x)$, encodes enumerative properties and allows the extraction of $\gamma_i(G)$ and $\Gamma_i(G)$ as the smallest and largest degrees with nonzero coefficients, respectively (Dod, 2016):

$$\gamma_i(G) = \min\{ k : d_k > 0 \}, \qquad \Gamma_i(G) = \max\{ k : d_k > 0 \}$$

Structural recurrences include, for vertex $v$:

$$D_i(G,x) = D_i(G-v, x) - D_i(G \circ v, x) + x D_i(G-N[v], x)$$

For edge $e = \{u,v\}$:

$$D_i(G,x) = D_i(G-e,x) - x^2 D_i(G-N[u,v],x) + x D_i((G \circ u)-N[v],x) + x D_i((G \circ v)-N[u],x)$$

Special graph families permit closed forms; for example:

• $D_i(K_n,x) = n x$
• $D_i(P_n,x) = x D_i(P_{n-2}, x) + x D_i(P_{n-3}, x)$ for $n \geq 4$

Generating function techniques yield direct formulas for the number of independent dominating sets of specified cardinality via binomial-möbius sums (Dod, 2016).

4. Optimization Formulations and Well-Coveredness

All maximal independent sets correspond to characteristic vectors in the binary solution set of a linear complementarity problem (LCP) with $M = A + I$, $q = -e$, where $A$ is the adjacency matrix:

$$\text{Find } x \geq 0,\; (A+I)x \geq e,\; x_i ((A+I)x - e)_i = 0 \; \forall i$$

The minimum $\ell_1$ norm over the solution set, $m(G) = \min\{e^T x : x \in \Sol(A+I, -e)\}$, gives a lower bound on the independent domination number, and coincides with it exactly if $G$ is a forest (Pandit et al., 2016).

A graph is well-covered if all maximal independent sets have the same size; in this case, all integer LCP solutions have the same $\ell_1$ norm (Pandit et al., 2016).

The LCP and LPCC frameworks expose the strong inapproximability of the independent domination number: computing or even approximating $\gamma_i(G)$ is strongly NP-hard and not approximable within $n^{1-\epsilon}$ unless $P = NP$ (Pandit et al., 2016).

5. Independence-Domination Parameters and Algorithmic Complexity

The independence-domination number $\gamma_{\rm ind}(G)$ is defined as the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$:

$$\gamma_{\rm ind}(G) = \max_{I\,\text{independent}} \gamma_G(I)$$

with $$\gamma_G(I) = \min\{ |D| : D \subseteq V,\, I \subseteq N[D] \}$$ (Hon et al., 2013).

• $\gamma(G) \ge \gamma_{\rm ind}(G)$.
• For chordal graphs, $\gamma(G) = \gamma_{\rm ind}(G)$ (Hon et al., 2013, Aharoni et al., 2017).

The problem is NP-complete for general graphs, including bipartite and chordal graphs (Hon et al., 2013). However, polynomial-time algorithms exist for cographs ($\gamma_{\rm ind}(G) =$ number of connected components), distance-hereditary graphs, permutation graphs, and bounded treewidth graphs. For planar graphs, a Baker-style PTAS yields, for every $\varepsilon > 0$, a $(1+\varepsilon)$-approximation in linear time (Hon et al., 2013).

For bounded treewidth $k$, dynamic programming algorithms compute $\gamma_{\rm ind}(G)$ in $O(n^3)$ time (Hon et al., 2013).

6. Weighted and Fractional Extensions

Weighted independent domination numbers generalize classical definitions by considering a vertex-weight function $w$:

• Weighted domination: minimum total weight of a set $D$ dominating $V$.
• Weighted independent domination: maximum, over all independent sets $I$, of the minimal total weight to dominate $I$.

For interval graphs and line-graphs of single-edge subtree intersections, the weighted independent domination number equals the weighted domination number, and both can be computed by linear or near-linear time greedy algorithms (Aharoni et al., 2017). For chordal graphs, equality may fail with weights.

Greedy and dual-packing algorithms, exploiting properties such as the consecutive-ones property of interval matrices, are central in these exact weighted results (Aharoni et al., 2017).

7. Connections, Special Cases, and Open Questions

Maximal independent domination interacts closely with classic invariants such as the independence number $\alpha(G)$, domination number $\gamma(G)$, and structural graph decompositions. The study of the independence domination polynomial links enumerative combinatorics, split decompositions, and Möbius inversion techniques.

Open questions highlighted include the precise boundaries of graph classes where weighted independence-domination and domination coincide, the complexity in broader chordal classes, and structural characterization of extremal examples. Special families—cographs, interval graphs, certain line-graphs—admit complete answers (Aharoni et al., 2017, Dod, 2016, Hon et al., 2013).

The theory and algorithms for maximal independent domination, including bounds, optimization characterizations, enumeration, and parameterized complexity, underpin a substantial corpus in modern graph theory and combinatorial optimization (Cho et al., 2022, Pandit et al., 2016, Hon et al., 2013, Aharoni et al., 2017, Dod, 2016).