---
title: Connected Coalition in Graph Theory
url: https://www.emergentmind.com/topics/connected-coalition
type: topic
---

# Connected Coalition in Graph Theory

Searching arXiv for relevant papers on connected coalitions and graph-constrained coalition formation.
I’ll use the available arXiv search capability to ground the article in current literature.
Connected coalition is a graph-based notion whose meaning depends on the surrounding theory. In graph-constrained coalition formation, coalition structure generation, and interaction-graph cooperative games, a coalition is feasible or viable exactly when the induced subgraph is connected. In graph domination, by contrast, a connected coalition is a pair of disjoint vertex sets whose union is a connected dominating set, while neither set alone is a connected dominating set. In partially connected device-to-device networks, the same graph-induced connectivity condition is imposed on subsets of devices, and coalition formation is used to coordinate instantly decodable network coding transmissions [1612.04299][1102.1747][1502.07713][2302.05754][1911.05201].

## 1. Terminological scope and formal definitions

The literature uses *connected coalition* in more than one non-equivalent way. The common substrate is an undirected graph \(G\) whose vertices are agents, devices, or graph vertices; connectivity of an induced subgraph is then used either as a feasibility condition for a single coalition or as a property that emerges only after two non-dominating parts are combined [1612.04299][2302.05754].

| Setting | Connected coalition condition | Canonical object |
|---|---|---|
| Graph-constrained coalition formation | \(C \subseteq A\) is feasible iff \(G[C]\) is connected | coalition structure \(CS \in CS(G)\) |
| Interaction-graph cooperative games | \(S \subseteq V\) is viable iff \(G[S]\) is connected | core, covering LP, packing LP |
| Connected-domination theory | disjoint \(A_1,A_2\) with \(A_1 \cup A_2\) a connected dominating set, but neither \(A_1\) nor \(A_2\) alone is | connected coalition partition, \(CC(G)\) |
| Partially connected D2D networks | \(S \subseteq V\) is connected iff the induced subgraph \(G[S]\) is connected | coalition-formation game over disjoint connected coalitions |

For graph-constrained coalition formation, let \(A=\{a_1,\dots,a_n\}\) be the set of agents and \(G=(A,E)\) an undirected graph whose vertices are agents and whose edges encode allowable pairwise relationships. A subset \(C \subseteq A\) is feasible iff the induced subgraph \(G[C]\) is connected, and a coalition structure is a partition of \(A\) into disjoint feasible coalitions [1612.04299]. Equivalent formulations appear in coalition-structure generation over graphs, where a connected coalition structure \(P=\{C_1,\dots,C_k\}\) is any partition such that each \(C_i\) induces a connected subgraph of \(G\) [1410.6516][1102.1747].

For connected-domination theory, let \(G=(V,E)\) be a simple graph. Two disjoint subsets \(V_1,V_2 \subseteq V\) form a connected coalition if neither \(V_1\) nor \(V_2\) is a connected dominating set, yet \(V_1 \cup V_2\) is a connected dominating set. A connected coalition partition is then a vertex partition in which each part is either a singleton full vertex or forms such a coalition with another part; the connected coalition number \(CC(G)\) is the maximum cardinality of such a partition [2302.05754][2402.00590].

## 2. Connected coalitions as feasible sets on interaction graphs

In graph-constrained coalition formation, the basic optimization problem is to maximize the value of a partition subject to connectivity. With \(FC(G)\) denoting the family of feasible coalitions and \(CS(G)\) the set of all feasible coalition structures, a characteristic function \(v:FC(G)\to\mathbb{R}\) is extended to coalition structures by
\[
V(CS)=\sum_{C\in CS} v(C),
\]
and the objective is
\[
CS^* = \arg \max_{CS\in CS(G)} V(CS).
\]
This is the standard GCCF formulation [1612.04299].

The same connectivity restriction appears in coalition-structure generation over graphs. Given an undirected graph \(G=(N,E)\) and a valuation function \(v:2^N \to \mathbb{R}\), the problem is to find a partition of \(N\) into connected subsets that maximizes the sum of coalition values:
\[
\max_{\Pi \text{ a connected partition of }N}\;\sum_{C\in\Pi} v(C).
\]
Voice, Polukarov, and Jennings formalized this as Graph Coalition Structure Generation (GCSG) and analyzed it under valuation functions that are *independent of disconnected members (IDM)*, including the edge-sum subclass
\[
v(C)=\sum_{e\in E[C]} w_e.
\]
The IDM condition can be stated either through marginal contributions across vertex separators or, equivalently, by the additivity relation \(v(B\cup C)=v(B)+v(C)\) for disconnected coalitions \(B\) and \(C\) [1102.1747][1401.6428].

A closely related cooperative-game formulation uses Myerson’s connectivity viability condition: a coalition \(S\subseteq V\) is viable exactly when \(G[S]\) is connected, and \(v(S)=0\) if \(G[S]\) is disconnected. In this setting, the graph is an interaction graph constraining which coalitions can realize positive value [1502.07713]. This suggests that, across these literatures, connectivity acts not as an auxiliary regularizer but as the primitive feasibility condition that defines the admissible coalition space.

## 3. Connected coalitions in connected-domination theory

In graph theory, the connected-coalition literature studies a different object. A set \(D\subseteq V\) is a connected dominating set if \(G[D]\) is connected and every vertex in \(V\setminus D\) has a neighbor in \(D\). Two disjoint sets form a connected coalition when neither is a connected dominating set but their union is. A connected coalition partition \(\psi=\{V_1,\dots,V_k\}\) is valid when each part is either a singleton \(\{v\}\) with \(\deg(v)=n-1\) or a non-connected-dominating set that forms a connected coalition with another part. The connected coalition number \(CC(G)\) is the maximum number of parts in such a partition [2302.05754].

The early structural theory identifies the graphs that admit no connected coalition partition. If \(\mathcal F\) is the family obtained by including all disconnected graphs of order at least \(2\) and then repeatedly taking the join \(H+K_1\), then \(CC(G)=0\) if and only if \(G\in\mathcal F\). For connected graphs outside \(\mathcal F\), one has \(1\le CC(G)\le n\); equality \(CC(G)=1\) occurs only for \(G=K_1\). The same work proves that if \(\delta(G)=1\) and \(G\) has no full vertex, then \(CC(G)<n\), and that for any tree \(T\), \(CC(T)=2\) [2302.05754].

Subsequent work refines the theory. For a connected graph \(G\) of order \(n\ge 2\) with no full vertex, let
\[
X=\{\,v\in V(G)\mid G-v \text{ is disconnected}\,\}
\]
be the set of cut-vertices. Then
\[
CC(G)=2 \quad \Longleftrightarrow \quad X \text{ is a connected dominating set of } G.
\]
Exact values were also obtained for unicycle graphs, the corona product, and the join of two graphs, together with lower bounds for the Cartesian and lexicographic products [2402.00590]. For example, if \(G,H\) are any graphs, then
\[
CC(G\vee H)=
\begin{cases}
|V(G)|+|V(H)|, & \text{if neither }G\text{ nor }H\text{ is complete},\\
0, & \text{if one of }G,H\text{ is complete and the other has }CC=0,\\
CC(G)+CC(H), & \text{otherwise}.
\end{cases}
\]

The partition perspective also induces a secondary graph object. Given a connected coalition partition \(\pi(G)=\{V_1,\dots,V_k\}\), the coalition graph \(CCG(G,\pi)\) has vertices corresponding to the parts of \(\pi\), with an edge between \(V_i\) and \(V_j\) exactly when \(V_i\cup V_j\) is a connected dominating set in \(G\). For subcubic graphs, every such coalition graph is either a star \(S_k\) with \(k\ge 2\) or one of \(22\) explicitly listed small graphs of order at most \(6\); among the \(23\) resulting types, exactly nine are finitely realized by subcubic graphs [2509.04204].

A notable literature issue is that the reported values of \(CC(C_n)\) are not uniform across the supplied sources. The exposition of “Connected coalitions in graphs” gives \(CC(C_4)=2\) [2302.05754], whereas “On the connected coalition number” states
\[
CC(C_n)=
\begin{cases}
4,& n=4,\\
3,& n\ne 4.
\end{cases}
\]
[2402.00590]. This discrepancy should be read as a difference within the current literature rather than as a settled identity.

## 4. Computational complexity and algorithmic methods

Connectivity restrictions do not remove the essential hardness of coalition-structure generation. GCSG is NP-complete on general graphs even when the valuation is edge-sum, and it remains NP-complete on planar graphs; hence it is NP-complete on \(K_k\)-minor-free classes for \(k\ge 5\) [1102.1747][1401.6428]. Voice et al. also give positive results: on trees, GCSG is solvable in \(O(n^2)\), and on \(K_{2,3}\)-minor-free or \(K_4\)-minor-free graphs it is solvable in \(O(n^3)\) [1102.1747].

For bounded-treewidth graphs, dynamic programming yields exact linear-time algorithms in the number of vertices for fixed width. Given a tree decomposition of width \(\tau\), the tree-DP algorithm of Voice et al. solves the connected coalition-structure generation problem in time \(O(\tau^{\tau+O(1)}\cdot n)\). More generally, separator theorems lead to subexponential algorithms on sparse hereditary classes; for planar graphs the resulting bound is
\[
\exp\bigl(O(\sqrt n \log n)\bigr)
\]
[1401.6428].

An alternative exact approach for Graph-Constrained Coalition Formation is CFSS, which represents the search space through edge contraction on a 2-coloured graph and traverses a rooted CF-tree by depth-first branch-and-bound. CFSS is particularly efficient for characteristic functions of the form \(m+a\), where \(V(CS)=V^+(CS)+V^-(CS)\) with \(V^+\) superadditive and \(V^-\) subadditive. The method is anytime, admits a parallel variant P-CFSS, and, on the reported benchmarks, is up to \(10^4\times\) faster than DyCE on sparse networks, explores on average at most \(0.32\%\) or \(0.28\%\) of the full search tree in two benchmark families, and provides approximate solutions with quality guarantees for instances with more than \(2700\) agents [1612.04299].

Rahwan and Michalak develop a hybrid algorithm \(D\) that combines dynamic programming with depth-first tree search on a pseudotree. The worst-case time remains \(O(3^n)\) on a clique, matching DyPE, but the hybrid is empirically reported to significantly outperform both constituent parts when the subset-evaluation function has certain intuitive properties [1410.6516].

Recent work also formulates connected coalition formation as repeated graph splitting via QUBO. In GCS-Q, a connected weighted graph \(G=(V,E,w)\) is partitioned top-down, each split being accepted only if it increases the total sum of intra-coalition weights, and connectivity is guaranteed by construction because each accepted part is replaced by its connected components. In the reported LEO-satellite experiments, the D-Wave Advantage annealer significantly outperforms Gurobi in runtime while maintaining solution quality, and the full clustering pipeline yields a \(60\)–\(65\%\) reduction in total communication links in the Starlink instances listed in the paper [2408.06007].

## 5. Cooperative-game formulations on interaction graphs

When connected coalitions are treated as viable coalitions of a cooperative game, the central objects are the core, covering LPs, and packing LPs. Let \(G=(V,E)\) be an interaction graph, and let \(v:2^V\to\mathbb{R}_+\) be superadditive and satisfy the viability condition that \(v(S)>0\) only if \(G[S]\) is connected. A payoff vector \(x\in\mathbb{R}^V\) lies in the core if
\[
\sum_{i\in V} x_i = v(V)
\]
and
\[
\sum_{i\in S} x_i \ge v(S)
\quad \forall\, S\subseteq V \text{ with } G[S] \text{ connected.}
\]
Core non-emptiness is tested by a covering LP, and the dual is a fractional packing of viable coalitions [1502.07713].

The covering-packing duality induces several graph parameters. If \(p_f(G)\) is the value of the fractional packing LP and \(p(G)\) the maximum integral packing, then the primal integrality gap is \(\kappa(G)\), the dual integrality gap is \(\rho(G)\), and the packing-covering ratio is
\[
\Pi(G)=\kappa(G)\rho(G).
\]
For graphical coalition games, \(\Pi(G)\le \mathrm{tw}(G)+1\), where \(\mathrm{tw}(G)\) is the treewidth [1502.07713].

The same paper introduces the **thicket number** \(\tau(G)\), defined as the maximum hitting size over thickets, and the **vine-width** \(\nu(G)\), defined via vine decompositions. The central min-max theorem is
\[
\tau(G)=\nu(G).
\]
Moreover,
\[
\nu(G)\le \mathrm{tw}(G)+1
\quad\text{and}\quad
\mathrm{tw}(G)\le 2\nu(G)-1.
\]
The packing-covering ratio is characterized exactly by the thicket number: for every interaction graph \(G\), there exists a coalition game whose packing-covering ratio equals \(\tau(G)\), and every game on \(G\) satisfies \(\Pi(G)\le \tau(G)\). The worst-case primal gap satisfies
\[
\tau(G)/4 \le \max_{\text{games}} \kappa(G) \le \tau(G),
\]
while the worst-case dual gap satisfies
\[
c\cdot \tau(G)^\delta \le \max_{\text{games}} \rho(G) \le \tau(G)
\]
for some constants \(c,\delta>0\). The graphical set family of connected subsets also links the theory to VC-dimension: if \(d\) is the VC-dimension of \(\{S\subseteq V: G[S]\text{ connected}\}\), then \(\tau(G)\le d\) and therefore \(\Pi(G)\le d\) [1502.07713].

These results place connected coalitions inside a broader combinatorial framework. They show that once connectivity is encoded as viability, the main quantitative questions become covering, packing, integrality gaps, and decomposition width rather than only coalition enumeration.

## 6. Connected coalitions in partially connected D2D networks

In IDNC-assisted D2D communications, connected coalition appears in a directly operational form. The network is modeled by an undirected graph \(G=(V,E)\) over \(N\) wireless devices, where \((u,u')\in E\) if and only if \(u'\) lies in the coverage zone of \(u\) and vice versa. For any subset \(S\subseteq V\), the induced subgraph \(G[S]\) determines coalition feasibility: \(S\) is a connected coalition exactly when \(G[S]\) is connected, meaning that for every pair \(u,u'\in S\) there exists a path in \(G[S]\) linking them [1911.05201].

The original optimization is an NTU coalition game for D2D completion-time minimization. At transmission round \(t\), a coalition \(S\) may elect a subset \(A\subseteq S\) of simultaneous transmitters, each sending an XOR of packets \(\kappa_u\). With \(\mathcal W_u(t)\) the set of packets still wanted by user \(u\), \(\mathfrak D_u(t)\) the cumulative decoding delay, and
\[
T_u(t)\coloneqq \frac{|\mathcal W_u(0)|+\mathfrak D_u(t)-E[\sigma_u]}{1-E[\sigma_u]},
\]
the individual payoff is
\[
u_u(S)\coloneqq -T_u-[\mathfrak D_u(t)-\mathfrak D_u(t-1)].
\]
Because the coalition payoff is vector-valued and non-transferable, the coalition’s overall payoff is \(\phi(S)=(u_u(S))_{u\in S}\), and a scalar value function may be defined by
\[
v(S)\coloneqq \max_{u\in S} u_u(S).
\]
The grand-coalition version is intractable, and the paper states that finding the grand coalition and its optimal encoding policy is NP-hard [1911.05201].

The relaxation is a coalition-formation game over a partition \(\Psi=\{S_1,\dots,S_k\}\) of disjoint, connected coalitions. Each coalition selects one transmitting device \(a^*\in S\) and an IDNC packet \(\kappa_{a^*}\) by solving
\[
a^* = \arg \max_{a\in S}
\left[
|C_a\cap N_S|
+\sum_{u\in L_S\cap \tau(\kappa_a)} \log(1/\sigma_{a,u})
\right],
\]
where \(N_S\) are members of \(S\) still wanting packets, \(L_S\) are critical users, and \(\tau(\kappa_a)\) are the users targeted by \(\kappa_a\) [1911.05201].

Coalition updates are driven by Pareto preferences through classical merge-and-split rules. A collection \(\{S_1,\dots,S_m\}\) may merge into \(S=\cup_k S_k\) if at least one member strictly improves without hurting any other member and \(S\) remains connected. A coalition \(S\) may split into connected subcoalitions \(\{S_1,\dots,S_m\}\) if at least one member in each \(S_k\) strictly improves without hurting others and each \(S_k\) allows a non-zero number of targeted users. Starting from any initial partition \(\Psi_0\), successive merge/split steps lead in a finite number of steps to a final partition \(\Psi^*\) in which no further Pareto-improving move exists. The terminal partition is Nash-stable, and equivalently both \(D_{hp}\)-stable and \(D_c\)-stable [1911.05201].

In this application, connected coalition is neither only a graph-theoretic partition concept nor only an abstract feasibility condition. It is the local communication structure within which simultaneous transmissions, packet combinations, decoding delay, and convergence guarantees are jointly defined.

Source: https://www.emergentmind.com/topics/connected-coalition