Edge Coalition: Concepts & Applications

• Edge Coalition is a multi-disciplinary concept defined as cooperative structures in graphs where two disjoint, non-dominating edge sets together form an edge dominating set, with applications in graph partitioning.
• In network science, it denotes collectively influential edges whose coordinated removal triggers abrupt connectivity collapse, measured using metrics like edge collective influence (ECI).
• In edge computing, the term describes coalitions or federations among service providers and clients, leading to improved resource allocation and significant cost and energy savings.

Edge coalition is a polysemous term spanning at least three technical literatures. In graph theory, it denotes a domination-theoretic relation between two disjoint edge sets: neither set is edge dominating on its own, but their union is edge dominating (Mojdeh et al., 2023). In network science, closely related language is used for collectively influential edge sets whose coordinated removal induces abrupt dismantling transitions (Peng et al., 2023). In edge computing and mobile edge computing, the term is used more operationally for coalitions, federations, or coalition-formation games among edge providers, edge servers, clients, or miners rather than for subsets of graph edges (Cao et al., 2019).

1. Graph-theoretic definition and basic objects

In the graph-theoretic literature, the primitive notion is edge domination. For a graph $G=(V,E)$, an edge dominating set is a subset $D\subseteq E$ such that every edge not in $D$ is adjacent to at least one edge of $D$. On that basis, an edge coalition in $G$ consists of two disjoint sets of edges $E_1$ and $E_2$, neither of which is an edge dominating set, but whose union $E_1\cup E_2$ is an edge dominating set (Mojdeh et al., 2023).

The corresponding partition notion is an edge coalition partition, or ec-partition. If $\pi=\{E_1,\dots,E_k\}$ is an edge partition of $E(G)$, then each part must satisfy one of two conditions: it is a singleton edge dominating set, or it is not an edge dominating set but forms an edge coalition with another part of the partition. The maximum possible order of such a partition is the edge coalition number

$D\subseteq E$0

An ec-partition of order $D\subseteq E$1 is called an $D\subseteq E$2-partition (Mojdeh et al., 2023).

A closely related construction is the edge coalition graph $D\subseteq E$3. Its vertices correspond one-to-one with the parts of an ec-partition $D\subseteq E$4, and two vertices are adjacent exactly when the corresponding parts form an edge coalition in $D\subseteq E$5. This turns coalition compatibility inside $D\subseteq E$6 into an ordinary graph structure on the partition blocks (Besharati et al., 26 Jul 2025).

2. Foundational properties, examples, and local constraints

The foundational existence statement is strong: every graph has an ec-partition. One proof route uses the line graph $D\subseteq E$7: edge domination in $D\subseteq E$8 is equivalent to vertex domination in $D\subseteq E$9, so a coalition partition of $D$0 yields an edge coalition partition of $D$1. The same line-graph viewpoint also clarifies why edge coalitions are an edge-analogue of earlier vertex-coalition notions (Besharati et al., 26 Jul 2025).

The literature’s motivating examples are small but structurally revealing. For the path $D$2, the partition

$D$3

is an ec-partition, and $D$4. For the cycle $D$5, the singleton partition is already an ec-partition, so $D$6 (Mojdeh et al., 2023).

Several general constraints govern how large $D$7 can be. If $D$8 has no full edges, then no part in an ec-partition can itself be edge dominating; hence every part must participate in a coalition with some other part. If $D$9 has no full edge and minimum degree $D$0, then

$D$1

If $D$2 has $D$3 universal vertices, then

$D$4

At the local level, if $D$5 has maximum degree $D$6 and $D$7 is an $D$8-partition, then any part $D$9 can be in at most $G$0 edge coalitions (Besharati et al., 26 Jul 2025, Mojdeh et al., 2023).

3. Exact values, extremal cases, and structural classifications

For several standard graph families, the edge coalition number is known exactly. The 2025 note reports

$G$1

$G$2

and for paths and cycles,

$G$3

It also gives $G$4 and $G$5 as special cases of broader line-graph-based transfer results (Besharati et al., 26 Jul 2025).

Extremal classifications focus on the case $G$6, where $G$7, so the singleton partition achieves the maximum possible number of parts. For unicyclic connected graphs, this occurs exactly for an explicit family denoted $G$8 in the 2025 note; for trees, it occurs exactly for a family $G$9 consisting of trees with diameter at most $E_1$0, or diameter $E_1$1 with the middle vertex of any longest path having degree $E_1$2. For connected graphs that are neither trees nor unicyclic, the same note identifies a finite family $E_1$3 of special graphs for which $E_1$4 (Besharati et al., 26 Jul 2025).

The earlier 2023 paper develops parallel classifications. It proves, for example, that for connected graphs $E_1$5,

$E_1$6

$E_1$7

$E_1$8

and it gives characterizations of unicyclic graphs with $E_1$9, trees with $E_2$0, and broader connected graphs with $E_2$1 via explicit families $E_2$2, $E_2$3, and $E_2$4 (Mojdeh et al., 2023).

The small-value classifications are not entirely uniform across the two edge-coalition papers. The 2025 note states

$E_2$5

while retaining

$E_2$6

This suggests that exact small-case identifications and graph labels should be checked against the specific paper version being used (Besharati et al., 26 Jul 2025, Mojdeh et al., 2023).

The associated edge coalition graphs have also been classified for special families. For $E_2$7, the coalition graphs are limited to types such as $E_2$8, $E_2$9, and $E_1\cup E_2$0. For a star $E_1\cup E_2$1, the coalition graph under the singleton partition is

$E_1\cup E_2$2

Both papers also single out rare self-edge coalition graphs, although the internal graph labels reported for these cases differ across versions (Besharati et al., 26 Jul 2025, Mojdeh et al., 2023).

4. Relation to vertex coalitions and graph-constrained coalition formation

A recurrent source of terminological confusion is the distinction between edge coalitions and the connected coalition number studied in domination theory. In “On the connected coalition number,” the objects are vertex subsets, not edge subsets. There, a pair of vertex-disjoint sets $E_1\cup E_2$3 and $E_1\cup E_2$4 form a connected coalition when $E_1\cup E_2$5 is a connected dominating set, but neither $E_1\cup E_2$6 nor $E_1\cup E_2$7 is a connected dominating set. The paper defines connected coalition partitions, the invariant $E_1\cup E_2$8, characterizes graphs with $E_1\cup E_2$9, obtains exact values for unicycle graphs, gives formulas for corona products and joins, and proves lower bounds for Cartesian and lexicographic products. It is therefore a vertex-based connected analogue, not an edge-coalition theory in the literal sense (Guan et al., 2024).

A second adjacent literature is graph coalition structure generation. Here the coalition is still a subset of vertices, but coalitions must be connected in an underlying graph and are evaluated by a valuation function $\pi=\{E_1,\dots,E_k\}$0. Under the condition of independence of disconnected members (IDM), disconnected components contribute additively, so one may restrict attention to connected coalition structures. The edge-centric special case is the edge sum coalition valuation

$\pi=\{E_1,\dots,E_k\}$1

where coalition value is determined by the sum of weights of edges in the induced subgraph. This is the closest formalism in that paper to an “edge-based coalition,” but the coalition itself remains a vertex set. Computationally, the problem is generally NP-complete; with a known tree decomposition of width $\pi=\{E_1,\dots,E_k\}$2, it can be solved in

$\pi=\{E_1,\dots,E_k\}$3

hence in $\pi=\{E_1,\dots,E_k\}$4 for fixed $\pi=\{E_1,\dots,E_k\}$5; yet the edge-sum case remains NP-complete even for planar graphs, and a 3-SAT instance with $\pi=\{E_1,\dots,E_k\}$6 clauses can be represented by a planar instance with $\pi=\{E_1,\dots,E_k\}$7 nodes (Voice et al., 2014).

5. Collective edge behavior and explosive vulnerability

In network science, “edge coalition” is often used more interpretively to denote a collectively influential set of edges whose joint removal produces a sharp connectivity collapse. “Unveiling Explosive Vulnerability of Networks through Edge Collective Behavior” formulates the edge-based network dismantling problem as: given $\pi=\{E_1,\dots,E_k\}$8 and a target giant connected component size $\pi=\{E_1,\dots,E_k\}$9, find a minimum set of removed edges $E(G)$0 such that in the residual graph $E(G)$1,

$E(G)$2

The central phenomenon is explosive vulnerability: connectivity appears stable for many removals and then fails abruptly (Peng et al., 2023).

The paper develops edge collective influence (ECI) by mapping the original graph to its line graph. If $E(G)$3 is the edge-retention vector, with removed-edge fraction

$E(G)$4

then the dismantling problem is tied to minimizing a spectral stability quantity

$E(G)$5

with threshold $E(G)$6 determined by $E(G)$7. The finite-radius surrogate cost is

$E(G)$8

which yields the score

$E(G)$9

The resulting greedy decimation procedure repeatedly removes the edge with the largest score; for $D\subseteq E$00, the reported complexity is about $D\subseteq E$01 when removing a finite proportion of edges each round (Peng et al., 2023).

Two improved variants refine the removal profile. IECI targets hidden dismantling by reinserting or merging components according to

$D\subseteq E$02

whereas IECIR reorders the same edge set for fast dismantling using

$D\subseteq E$03

The paper further introduces dual competitive percolation (DCP) and IDCP as forward growth models whose reverse processes reproduce the explosive dismantling trajectory and the cost-function evolution of ECI. On synthetic networks, including Erdős–Rényi and Barabási–Albert graphs, and on empirical networks such as a power grid, a collaboration network, an email network, and a social network, ECI and its variants are reported to outperform baselines including BG, EB, CI, EI, and GNDR (Peng et al., 2023).

6. Coalitions at the network edge: federation, tolling, learning, and blockchain

In edge-computing research, the phrase “edge coalition” usually does not refer to graph edges. Instead, it denotes coalitions or federations among edge infrastructure providers, edge servers, clients, or miners. “Edge Federation: Towards an Integrated Service Provisioning Model” treats multiple edge infrastructure providers and clouds as a trusted consortium with horizontal integration among edge nodes and vertical integration between edge and cloud. Service provisioning is formulated as a large-scale linear program with fractional storage and computation assignment variables, cost objective $D\subseteq E$04, and latency and satisfaction constraints. The paper proposes the dynamic algorithm SEE (Service provision for Edge fEderation) and reports cost reductions of $D\subseteq E$05 and $D\subseteq E$06 relative to the fixed contract model in 30-node and 50-node cases, and $D\subseteq E$07 and $D\subseteq E$08 relative to the multihoming model; with 30, 60, and 120 minute slots, the reported savings are $D\subseteq E$09, $D\subseteq E$10, and $D\subseteq E$11, respectively (Cao et al., 2019).

“EdgeToll: A Blockchain-based Toll Collection System for Public Sharing of Heterogeneous Edges” operationalizes coalition-like cooperation among distinct service providers through blockchain settlement and payment channels. Its architecture includes edge nodes, end users, a proxy, and an Ethereum smart contract supporting payment channel establishment, signature verification, payment channel closure, and collateral query. The proxy matches users to edges and manages channels so that users need not open a channel to every edge individually. The implementation fixes the proxy’s edge-registration deposit at 1 ether and evaluates a testbed with 3 edge nodes, 1 terminal user device, Truffle Suite, Rinkeby Testnet, Solidity, web3.py, and Python. Tasks vary from 1 to 50, each repeated 100 times. The paper reports that for 50 tasks at 7 Gwei gas price, the no-channel system cost is about 0.0057 ether, about 345% larger than EdgeToll, and that in one scenario with 20 tasks and 20 edges, the saved cost is around 1.14 ether, described as nearly 93.7% reduction compared with a traditional system (Xiao et al., 2019).

In hierarchical federated learning, coalition formation appears at the level of client-to-edge-server association. “LEAP: Optimization Hierarchical Federated Learning on Non-IID Data with Coalition Formation Game” models the client set $D\subseteq E$12, edge server set $D\subseteq E$13, and coalition partition $D\subseteq E$14 with disjoint coalitions

$D\subseteq E$15

The objective couples execution time $D\subseteq E$16, energy $D\subseteq E$17, and average cross-edge Jensen–Shannon divergence $D\subseteq E$18, with utility

$D\subseteq E$19

Coalition switching is driven by JSD reduction, and the paper proves that the game is an exact potential game with at least one pure-strategy Nash equilibrium. After coalition formation, bandwidth is optimized by projected gradient descent and transmission power by a closed-form rule under delay constraints. On four real datasets, the abstract reports 20.62% improvement in model accuracy compared to state-of-the-art baselines and at least about 2.24 times lower transmission energy consumption; the switching example shows $D\subseteq E$20 decreasing from 0.69 to 0.49 and finally to 0.0 (Lu et al., 2024).

In MEC-assisted blockchain networks, coalition structure is made explicitly overlapping. “An Overlapping Coalition Game Approach for Collaborative Block Mining and Edge Task Offloading in MEC-assisted Blockchain Networks” lets a miner join multiple coalitions at once through membership variables $D\subseteq E$21, so that

$D\subseteq E$22

The system is modeled as a two-stage Stackelberg game: in Stage I, the edge computing service provider sets price $D\subseteq E$23; in Stage II, miners form overlapping coalitions and each coalition chooses how many edge resources to purchase. The paper derives a closed-form Nash equilibrium for the edge resource competition game, proves convergence of repeated atomic coalition updates to a stable coalition structure, and reports that the multi-coalition mode improves system efficiency by about $D\subseteq E$24 over the traditional single-coalition mode (Ye et al., 8 Aug 2025).

Taken together, these literatures show that “edge coalition” has no single universal meaning. In graph theory it is a precise edge-domination construct; in network dismantling it denotes collectively influential edge sets; and in edge computing it designates coalition or federation mechanisms among edge-side entities. The shared idea is cooperative structure under constrained interaction, but the underlying mathematical objects and performance criteria are domain-specific.