Connected Coalition Partition

• Connected Coalition Partition is a graph-based concept that partitions vertices into connected blocks to maximize coalition values or satisfy domination criteria.
• The approach employs dynamic programming, decomposition, and greedy heuristics to tackle NP-complete cases while achieving polynomial-time solutions on trees and minor-free graphs.
• Research leverages edge-sum valuations, separator theorems, and coalition graph structures to derive complexity bounds and optimize partitioning in various graph classes.

Connected Coalition Partition denotes graph-constrained partitioning under a connectivity requirement, but the literature uses closely related terminology for two distinct objects. In coalition structure generation over graphs, the task is to partition the vertex set into connected blocks that maximize an additive objective over coalition values. In connected-domination theory, a connected coalition partition is a partition whose parts are either single full vertices or pairs of sets whose union is a connected dominating set although neither set alone is connected dominating; the associated invariant is the connected coalition number $CC(G)$ (Voice et al., 2011, Alikhani et al., 2023). Both lines of work study how graph structure restricts feasible coalitions, but they optimize different quantities and employ different structural tools.

1. Optimization formulation on graphs

Let $G=(N,E)$ be an undirected graph with node set $N$, $|N|=n$, and let $v:2^N\to\mathbb{R}$ satisfy $v(\emptyset)=0$. A coalition structure, or partition, $\Pi=\{C_1,\dots,C_k\}$ is a collection of disjoint subsets whose union is $N$. It is connected if every induced subgraph $G[C_i]$ is connected. The Connected Coalition Partition problem is to find

$$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$

where $G=(N,E)$0 ranges over all connected partitions of $G=(N,E)$1 (Voice et al., 2011).

The same optimization problem appears in later work under the names coalition structure generation on graphs and Graph-Constrained Coalition Formation. In that formulation, a coalition $G=(N,E)$2 is feasible iff the induced subgraph $G=(N,E)$3 is connected, $G=(N,E)$4 denotes the set of all such connected partitions, and the value of a coalition structure is

$G=(N,E)$5

The goal is

$G=(N,E)$6

A dynamic-programming formulation also introduces $G=(N,E)$7 and

$G=(N,E)$8

with minimization variants obtained by replacing $G=(N,E)$9 with $N$0 (Bistaffa et al., 2016, Rahwan et al., 2014).

This formulation generalizes ordinary coalition structure generation by imposing graph feasibility. In complete graphs the connectivity constraint is vacuous, whereas in sparse graphs the admissible partition family can be much smaller.

2. Valuation classes and the complexity landscape

A central structural assumption is independence of disconnected members (IDM). The valuation $N$1 is IDM if whenever $N$2 are separated by a vertex separator $N$3,

$N$4

Intuitively, two nodes in different connected components of $N$5 do not affect each other’s marginal contribution beyond $N$6. An important corollary is that if $N$7 and $N$8 are disconnected subsets, then

$N$9

A particularly natural subclass is edge-sum valuation: $|N|=n$0 where each edge has weight $|N|=n$1 (Voice et al., 2011).

The basic complexity picture is heterogeneous. The problem is generally NP-complete, and this remains true even for edge-sum valuations. Hardness is obtained by reduction from 3-SAT using gadgets with a central node $|N|=n$2, literal nodes, edges of weight $|N|=n$3, and incompatibility edges of weight $|N|=n$4; any optimal connected partition must avoid negative edges and thereby encode a satisfying assignment if one exists. A planar hardness result is also known: edge-sum coalition structure generation on planar graphs is NP-complete, and a 3-SAT problem with $|N|=n$5 clauses can be represented by a planar instance with $|N|=n$6 nodes (Voice et al., 2011, Voice et al., 2014).

Graph class	Exact status	Bound
General graphs	NP-complete, even edge-sum	—
Trees	Polynomial-time solvable	$|N|=n$7
$|N|=n$8-minor-free graphs	Polynomial-time solvable	$|N|=n$9
$v:2^N\to\mathbb{R}$0-minor-free graphs	Polynomial-time solvable	$v:2^N\to\mathbb{R}$1
Bounded-treewidth graphs	Fixed-parameter exact DP	$v:2^N\to\mathbb{R}$2
$v:2^N\to\mathbb{R}$3-minor-free graphs	Subexponential exact	$v:2^N\to\mathbb{R}$4 for suitable $v:2^N\to\mathbb{R}$5
Planar graphs	Subexponential exact but NP-complete	$v:2^N\to\mathbb{R}$6 for suitable $v:2^N\to\mathbb{R}$7

For $v:2^N\to\mathbb{R}$8-minor-free graphs with $v:2^N\to\mathbb{R}$9, the bound is $v(\emptyset)=0$0 for any $v(\emptyset)=0$1; for planar graphs, $v(\emptyset)=0$2 for any $v(\emptyset)=0$3. When a tree decomposition of width $v(\emptyset)=0$4 is given, the problem can be solved in $v(\emptyset)=0$5, and hence in $v(\emptyset)=0$6 for fixed $v(\emptyset)=0$7 (Voice et al., 2011, Voice et al., 2014).

3. Exact, hybrid, and anytime algorithms

For tractable graph classes, the constructive proofs yield explicit algorithms. On trees, a leaf-peeling dynamic update repeatedly removes a leaf $v(\emptyset)=0$8 with neighbor $v(\emptyset)=0$9, computes $\Pi=\{C_1,\dots,C_k\}$0, recurses on $\Pi=\{C_1,\dots,C_k\}$1, and then either merges $\Pi=\{C_1,\dots,C_k\}$2 into the block containing $\Pi=\{C_1,\dots,C_k\}$3 if $\Pi=\{C_1,\dots,C_k\}$4 or keeps $\Pi=\{C_1,\dots,C_k\}$5 as a singleton. This runs in $\Pi=\{C_1,\dots,C_k\}$6. For 2-connected $\Pi=\{C_1,\dots,C_k\}$7-minor-free graphs, one exploits a simple cycle with at most two high-degree vertices and no chords, case-splits over how to partition that cycle, contracts or deletes paths, recurses on a smaller graph, and lifts the solution back, giving $\Pi=\{C_1,\dots,C_k\}$8. For $\Pi=\{C_1,\dots,C_k\}$9-minor-free graphs, the decomposition into 2-connected components reduces the remaining work to either $N$0-minor-free pieces or a $N$1 subgraph; since CCP on $N$2 is constant-time solvable by brute-force over its 15 partitions, the global runtime is again $N$3. For general $N$4-minor-free and planar graphs, separator theorems drive divide-and-conquer by enumerating connected partitions of the separator and stitching solutions via the IDM property (Voice et al., 2011).

For arbitrary connected graphs, dynamic programming and search-based exact methods were developed around pseudotree orderings. DyPE computes

$N$5

with $N$6, and in the worst case requires $N$7 time and $N$8 space. TSP performs depth-first search over connected partitions, maintaining a partial partition $N$9, branching on connected subsets containing the first unassigned vertex in breadth-first order, and pruning with an upper bound $G[C_i]$0. The hybrid algorithm $G[C_i]$1 runs a modified DyPE and a modified TSP in lock-step from opposite ends of the pseudotree ordering; DyPE supplies solved suffix subproblems, while TSP uses them to prune or instantly complete suffixes. The result is an anytime exact algorithm whose final incumbent is optimal when the two traversals meet (Rahwan et al., 2014).

A different representation underlies CFSS for Graph-Constrained Coalition Formation. Every connected partition can be generated by contracting a carefully chosen subset of edges in a 2-coloured graph $G[C_i]$2, where green edges remain contractible and red edges are forbidden from further contraction. Contracting a green edge $G[C_i]$3 produces the child partition

$G[C_i]$4

and marking $G[C_i]$5 red in the parent ensures that each partition arises exactly once. For $G[C_i]$6 functions, the branch-and-bound upper bound is

$G[C_i]$7

where $G[C_i]$8 is the coarsest partition obtained by removing all red edges and contracting every remaining green edge. If $G[C_i]$9 is the current frontier, then

$$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$0

is non-increasing and always bounds the true optimum from above; after any time budget, the approximate solution $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$1 has performance ratio $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$2. Empirically, the serial version of CFSS is 4 orders of magnitude faster than the state of the art in the best case, the parallel version is 9.44 times faster than the serial version on a 12-core machine, and the method is reported as the first GCCF approach to provide anytime approximate solutions with quality guarantees for very large systems of agents, with more than 2700 agents (Bistaffa et al., 2016).

4. Connected coalition partitions in domination theory

A different definition begins with domination rather than additive coalition value. For a finite simple graph $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$3, a set $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$4 is a dominating set if every vertex in $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$5 has a neighbor in $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$6, and it is connected dominating if $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$7 is connected. Two vertex-disjoint sets $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$8 form a connected coalition if $$\Pi^*=\arg\max_{\Pi\vdash_{\mathrm{conn}} N}\sum_{C\in\Pi} v(C),$$9 is a connected dominating set of $G=(N,E)$00, but neither $G=(N,E)$01 nor $G=(N,E)$02 alone is a connected dominating set. A connected coalition partition $G=(N,E)$03 is a partition of $G=(N,E)$04 such that each $G=(N,E)$05 is either a singleton $G=(N,E)$06 where $G=(N,E)$07 is a full vertex of degree $G=(N,E)$08, or there exists $G=(N,E)$09, $G=(N,E)$10, such that $G=(N,E)$11 form a connected coalition. The connected coalition number is

$G=(N,E)$12

(Alikhani et al., 2023, Guan et al., 2024).

This theory includes several extremal characterizations. If $G=(N,E)$13 is disconnected and $G=(N,E)$14, then $G=(N,E)$15. The only graph with $G=(N,E)$16 is $G=(N,E)$17. If $G=(N,E)$18 is the closure under join-with-$G=(N,E)$19 of all disconnected graphs of order at least $G=(N,E)$20, then

$G=(N,E)$21

For connected graphs of order $G=(N,E)$22 with $G=(N,E)$23 and no full vertex, $G=(N,E)$24. For any nontrivial tree $G=(N,E)$25,

$G=(N,E)$26

More precisely, if $G=(N,E)$27 is connected with no full vertices and

$G=(N,E)$28

then

$G=(N,E)$29

(Alikhani et al., 2023, Guan et al., 2024).

Several standard graph classes admit exact values. Every complete graph $G=(N,E)$30 satisfies $G=(N,E)$31 because every vertex is full. For complete bipartite $G=(N,E)$32 with $G=(N,E)$33, every pair consisting of one vertex from each side forms a connected coalition, so $G=(N,E)$34. For paths $G=(N,E)$35 and, more generally, any nontrivial tree without full vertex, $G=(N,E)$36. For cycles, $G=(N,E)$37, and for $G=(N,E)$38, $G=(N,E)$39. For a star $G=(N,E)$40, the center is full, and $G=(N,E)$41 (Alikhani et al., 2023, Guan et al., 2024).

5. Exact values under graph operations and coalition graphs

The connected coalition number behaves regularly on several graph operations. For a unicycle graph $G=(N,E)$42 of order $G=(N,E)$43 with cycle $G=(N,E)$44, let

$G=(N,E)$45

If $G=(N,E)$46 is the family obtained by identifying a vertex of $G=(N,E)$47 with the center of some star, then

$G=(N,E)$48

For the corona product,

$G=(N,E)$49

and for the join,

$G=(N,E)$50

If $G=(N,E)$51 and $G=(N,E)$52 denote the numbers of full vertices in connected graphs $G=(N,E)$53 and $G=(N,E)$54, then

$G=(N,E)$55

and for the lexicographic product,

$G=(N,E)$56

(Guan et al., 2024).

A further refinement is the coalition graph of a connected coalition partition. Given $G=(N,E)$57 and a partition $G=(N,E)$58, the coalition graph $G=(N,E)$59 has vertex set $G=(N,E)$60, and $G=(N,E)$61 exactly when $G=(N,E)$62 and $G=(N,E)$63 form a connected coalition in $G=(N,E)$64. For connected subcubic graphs, the characterization is sharp: $G=(N,E)$65 occurs as $G=(N,E)$66 for some connected subcubic $G=(N,E)$67 iff $G=(N,E)$68 is a star $G=(N,E)$69 with $G=(N,E)$70, or one of the following 22 small graphs of order at most $G=(N,E)$71: $G=(N,E)$72 Exactly nine of these 22 graphs can arise only from finitely many subcubic $G=(N,E)$73, namely

$G=(N,E)$74

whereas the remaining 13 admit infinitely many realizations. The same work proves that for every connected subcubic graph $G=(N,E)$75 of order $G=(N,E)$76,

$G=(N,E)$77

and for $G=(N,E)$78 the bound is sharp (Dobrynin et al., 4 Sep 2025).

6. Related formulations, applications, and open directions

A recurrent source of ambiguity is that connected coalition partition may denote either value-maximizing connected coalition structure generation or the domination-based parameter $G=(N,E)$79. A plausible implication is that algorithmic results do not transfer automatically between the two settings: one optimizes $G=(N,E)$80 over connected induced subgraphs, whereas the other maximizes the number of parts subject to connected-dominating coalition rules.

The optimization viewpoint has motivated application-specific formulations. In signed graphs, one model partitions $G=(N,E)$81 into $G=(N,E)$82 connected coalitions so as to minimize the total number of intra-group negative and inter-group positive edges. Connectivity is enforced by binary assignment variables $G=(N,E)$83, tree-edge selectors $G=(N,E)$84, and single-commodity flow variables $G=(N,E)$85 satisfying flow-balance and spanning-tree constraints. The decision version generalizes both the classical Min-Cut and Minimum Frustration/Line-Index of Balance problems. Computationally, the reported average solve time is 4 s for Senate networks, about 5 h per House session on average, and about 12 h for the densest 113th House instance; the application links increasingly polarized coalitions to legislative effectiveness in the House (Aref et al., 2019).

For GCCF proper, the practical message is mixed. If the underlying network is a tree or excludes small minors, exact solution is available in low-order polynomial time. Planar or general minor-free topologies admit $G=(N,E)$86-time algorithms, which can solve moderately large instances exactly. On general or planar graphs with arbitrary IDM valuations, even edge-sum, the problem is NP-complete; for large dense networks, exact solution is intractable, and the literature explicitly points to approximation or heuristics such as greedy cluster growth, local search on partitions, LP-based rounding under edge-sum weights, and exploitation of treewidth or community structure (Voice et al., 2011).

Open problems remain in the domination-based line as well. For coalition graphs of subcubic graphs, no general polynomial-time algorithm or NP-hardness result is given, and recognition of whether a given $G=(N,E)$87 is a coalition graph of some subcubic $G=(N,E)$88 remains open. The same work also shows that there is no single universal subcubic graph whose coalition-graph-set coincides with the entire class of coalition graphs of subcubic graphs (Dobrynin et al., 4 Sep 2025). Across both traditions, the subject exhibits a characteristic split: exact structure is strong on restricted graph classes, while general instances quickly reach NP-hardness.