---
title: 'Coalition Graph: Theory and Applications'
url: https://www.emergentmind.com/topics/coalition-graph
type: topic
---

# Coalition Graph: Theory and Applications

In contemporary graph theory, **coalition graph** denotes two different but technically well-defined objects. In domination-theoretic usage, it is an auxiliary graph attached to a **coalition partition**: vertices represent partition classes, and adjacency records which classes are coalition partners. In a separate line of research related to perfect orderability, a **coalition graph** is a graph admitting an acyclic orientation in which the two end-edges of every induced path on four vertices are directed the same way. The shared terminology reflects a common emphasis on coordinated local structure, but the two notions are formally independent [2511.21112] [1507.00557].

## 1. Domination-theoretic coalition graphs

Let \(G\) be a graph with vertex partition \(\pi=\{V_1,\dots,V_k\}\). A **coalition** consists of two disjoint sets \(V_i,V_j\) such that neither set is a dominating set, but \(V_i\cup V_j\) is a dominating set. A **coalition partition** (or **\(c\)-partition**) is a partition in which every part is either a singleton dominating set or a non-dominating set that forms a coalition with another part. The maximum order of such a partition is the **coalition number** \(C(G)\). Given a \(c\)-partition \(\pi\), the **coalition graph** \(CG(G,\pi)\) has vertex set \(\{V_1,\dots,V_k\}\), with \(V_iV_j\in E(CG(G,\pi))\) if and only if \(V_i\) and \(V_j\) are coalition partners in \(G\) [2511.21112].

This construction separates two levels of structure. The underlying graph \(G\) carries domination data, while \(CG(G,\pi)\) records how the partition classes cooperate to achieve domination. The same graph \(G\) may admit multiple \(c\)-partitions, so the associated coalition graph is generally not unique. Haynes et al. also defined the **coalition count** \(c(G)\) as the maximum number of different coalitions in any \(c\)-partition, which measures the density of coalition relations rather than the number of partition blocks [2511.21112].

A special case is the **singleton coalition graph**. If the singleton partition \(\Gamma_1=\{\{v\}:v\in V(G)\}\) is itself a coalition partition, then \(G\) is a **singleton-partition graph**, and \(CG(G,\Gamma_1)\) is its **singleton coalition graph**. This regime is especially useful for iterated constructions and graph-chain phenomena, because coalition adjacencies are then computed directly from vertex pairs of the original graph [2304.07606].

## 2. Realization theory and explicit constructions

A central structural result is a universality theorem: for every graph \(G\), there exist a graph \(H\) and a \(c\)-partition \(\pi\) such that \(CG(H,\pi)\cong G\). Swathi Shetty, Sayinath Udupa N. V., and B. R. Rakshith strengthened this by constructing a graph \(H^*\) of smaller order and size together with a \(c\)-partition \(\pi^*\) such that \(CG(H^*,\pi^*)\cong G\) [2511.21112].

The construction starts from a clique on the non-isolated vertices of \(G\), deletes selected edges to prevent singleton dominating sets, adds gadgets for non-edges, and treats isolates as dominating vertices. The resulting partition is initialized by singleton parts and extended as vertices are added. Two explicit properties are emphasized: none of the parts in \(\pi^*\) are dominating sets by themselves, and the coalition graph \(CG(H^*,\pi^*)\) encodes \(G\) exactly [2511.21112].

| Input graph \(G\) | \(n(H^*)\) | \(m(H^*)\) |
|---|---:|---:|
| \(K_n \cup tK_1\), \(n\) even | \(n+t\) | \(\binom{n}{2}-\frac{n}{2}+t(n+t-1)\) |
| \(K_n \cup tK_1\), \(n\) odd | \(n+t+1\) | \(\binom{n}{2}-\frac{n-1}{2}+n-2+t(n+t)\) |
| \(G' \cup tK_1,\ G' \not\cong K_n\) | \(n+\overline{m}_{G'}+t\) | see paper |

The singleton setting yields further realization phenomena. For example, the singleton coalition graph chain beginning with \(C_4\) is
\[
C_4 \rightarrow K_4 \rightarrow \overline{K}_4,
\]
while for \(P_3\) the chain alternates between \(P_3\) and \(K_1\cup K_2\). This shows that repeated passage to a singleton coalition graph may stabilize, cycle, or collapse to sparse limiting forms, depending on the starting graph [2304.07606].

## 3. Coalition number, coalition count, and extremal behavior

The invariants \(C(G)\) and \(c(G)\) capture different aspects of coalition structure and are not comparable in general. The complete graph satisfies \(C(K_n)=n\) and \(c(K_n)=0\), whereas for the 4-cycle one has \(C(C_4)=4\) and \(c(C_4)=6\). Thus maximal partition size need not imply many coalition edges, and a partition with relatively few blocks can still induce a dense coalition graph [2511.21112].

For graphs with no isolated vertices and \(f\) full vertices, the coalition count satisfies
\[
c(G)\ge d(G)-f,
\]
where \(d(G)\) is the domatic number; this bound is sharp for stars and complete graphs. A full characterization is available for the smallest positive value:
\[
c(G)=1 \iff \alpha(G)=n-f,
\]
where \(\alpha(G)\) is the independence number. Equivalently, all such graphs are isomorphic to \((K_f+pK_1)\cup qK_1\) for appropriate \(f,p,q\) [2511.21112].

Coalition graphs arising from special partition types can also be constrained quite rigidly. If a graph has exactly one full vertex \(u_1\), minimum degree \(1\), and coalition number \(C(G)=s\ge 2\), then \(c(G)=s-2\), and the coalition graph of a suitable \(C(G)\)-partition is
\[
CG(G,\pi)\cong K_1\cup K_{1,s-2}.
\]
For singleton-partition graphs with no full vertices, the bound \(c(G)\ge \alpha(G)\) further links coalition density to independence structure [2511.21112].

These results indicate that coalition graphs are best regarded as partition-dependent encodings of domination synergy, rather than as direct surrogates for the original graph. A plausible implication is that inverse problems—recovering \(G\) or identifying all \(\pi\) with a prescribed \(CG(G,\pi)\)—are intrinsically underdetermined unless additional structural constraints are imposed.

## 4. Variants generated by alternative domination notions

The domination-theoretic framework has been extended by replacing ordinary domination with stronger or modified notions. The associated auxiliary graphs preserve the same basic pattern: partition classes become vertices, and adjacency records when two classes jointly realize the target domination property.

| Variant | Partition notion | Associated graph |
|---|---|---|
| Paired domination | \(pc\)-partition | \(PCG(G,\pi)\) |
| Connected domination | connected coalition partition | \(CCG(G,\pi)\) |
| Secure domination | \(\sec\)-partition | \(SCG(G,\pi)\) |
| Edge domination | \(ec\)-partition of \(E(G)\) | \(ECG(G,\pi)\) |

For **paired coalitions**, \(PCG(G,\pi)\) is defined from partitions in which no part is a paired dominating set but every part has a paired coalition partner. For paths, the paired coalition graphs are exactly \(P_2\) and \(P_3\). For cycles, the possible paired coalition graphs are \(P_3\), \(K_3\), \(K_2\cup K_2\), \(P_4\), and \(C_4\). For trees with \(PC(T)\ge 3\), the paired coalition graph is always a star \(K_{1,k-1}\) [2402.10842].

For **connected coalitions**, the graph \(CCG(G,\pi)\) records partnerships whose unions are connected dominating sets. In subcubic graphs, the coalition graphs are completely characterized: they consist of the infinite family of stars \(S_k\) for \(k\ge 2\) together with 22 finite graphs of order at most 6, including \(K_1\), \(\overline{K}_2\), \(C_4\), \(K_{2,3}\), and \(K_{3,3}\). In the earlier foundational study of connected coalitions, all trees were shown to satisfy \(CC(T)=2\), and polynomial-time algorithms were given to determine whether \(CC(G)=n\) or \(CC(G)=n-1\) [2509.04204] [2302.05754].

For **secure coalitions**, every graph admits a secure coalition partition, and every graph without isolated vertices is a secure coalition graph: for every such graph \(G\), there exist \(H\) and \(\pi\) with \(SCG(H,\pi)\cong G\). This is a secure-domination analogue of the universality phenomenon for ordinary coalition graphs [2511.21170].

For **edge coalitions**, the partition is over \(E(G)\) rather than \(V(G)\). The edge coalition graph \(ECG(G,\pi)\) has partition classes of edges as vertices. In this setting, \(EC(G)=1\) if and only if \(G=K_2\), \(EC(G)=2\) if and only if \(G\in\{P_3,\overline{C_4}\}\), and \(EC(G)=3\) if and only if \(G\in\{C_3,P_4,K_{1,3}\}\). For stars \(K_{1,n-1}\), the only \(ec\)-partition is the singleton partition, and the edge coalition graph is \((n-1)K_1\) [2507.19871].

A noteworthy boundary case is **perfect coalition**. The theory of perfect coalitions and perfect coalition partitions has been initiated, together with the perfect coalition number \(\mathrm{PRC}(G)\), and every \(prc\)-partition is a coalition partition, so \(\mathrm{PRC}(G)\le C(G)\). However, the supplied data do not define an associated “perfect coalition graph”; current results emphasize partition existence, extremal values, and structural characterizations instead [2409.10185].

## 5. Coalition graphs as an orientation class

A different usage defines a **coalition graph** as a graph admitting an acyclic orientation such that, for every induced path \(P_4: abcd\), the end-edges \(ab\) and \(cd\) are oriented in the same direction. Such an orientation is called a **coalition orientation**. Equivalently, a graph is a coalition graph if and only if it admits a linear order \(<\) on its vertex set such that every induced \(P_4: abcd\) satisfies \(a<b\) if and only if \(c<d\). In some literature these are called **one-in-one-out graphs** [1507.00557].

This notion belongs to the theory of graph orientations and perfect orderability, not to domination partitions. All comparability graphs are coalition graphs, and hence all bipartite graphs are coalition graphs. The relationship with opposition graphs is parallel but reversed: opposition graphs require the two end-edges of each induced \(P_4\) to be oriented in opposition rather than in the same direction [1507.00557].

Recognition is open in general, but several restricted classes admit complete characterizations. For a graph \(G\), the auxiliary graph \(\mathscr{C}(G)\) has as vertices the ordered pairs \((x,y)\) where \(\{x,y\}\) is an end-edge of some induced \(P_4\); two such vertices are adjacent when one is the reverse of the other, or when they appear as the end-edges of an induced \(P_4\). If \(G\) is \((\mathrm{gem},\mathrm{house},\mathrm{hole})\)-free, then
\[
G \text{ is a coalition graph } \iff \mathscr{C}(G) \text{ is bipartite},
\]
yielding an \(O(m^2)\)-time recognition algorithm. For distance-hereditary graphs, the following are equivalent: \(G\) is a coalition graph, \(\mathscr{C}(G)\) is bipartite, \(G\) is \(\mathsf{N}\)-free, and \(G\) is a comparability graph; recognition and orientation are then possible in \(O(n+m)\) time [1507.00557].

If the acyclicity requirement is dropped, one obtains **generalized coalition graphs**. The supplied data note that, under this weaker definition, any cycle can be so oriented, and the resulting class is no longer a class of perfect graphs [1507.00557].

## 6. Relation to coalition formation over graphs

The term **coalition graph** should also be distinguished from the broader algorithmic literature on **coalition formation over graphs**, where a graph constrains which coalitions are feasible. In **Graph Coalition Structure Generation**, one is given an undirected graph \(G=(N,E)\) and a valuation \(v:2^N\to\mathbb{R}\), and seeks a partition of \(N\) into connected subsets maximizing \(\sum_{C}v(C)\). This optimization problem is NP-complete in general and remains NP-complete for planar graphs even for edge-sum valuations; polynomial-time algorithms are known for trees and for \(K_{2,3}\)- and \(K_4\)-minor-free graphs [1102.1747].

In **Weighted Graph Games**, the value of a coalition is the sum of edge weights in its induced subgraph. Finding the optimal coalition structure is NP-hard even on planar graphs, but approximation algorithms are available: an 18-approximation for planar graphs, an \(O(h^2\log h)\)-approximation for \(H\)-minor-free graphs, and a \((\Delta+1)\)-approximation for bounded-degree graphs [1108.5248].

In **Graph-Constrained Coalition Formation**, only connected coalitions are feasible; CFSS represents the search space by edge contraction, yielding a rooted tree in which each node uniquely represents a feasible coalition structure, with \(O(|\mathcal{CS}(G)|\cdot |\mathcal{E}|)\) node-processing complexity. On reported benchmarks, the serial version is up to 4 orders of magnitude faster than the state of the art, and the parallel version achieves a 9.44x speedup on a 12-core machine [1612.04299].

A further distinction arises in **graph hedonic games**, where players move among connected coalitions according to preference relations. Individually stable dynamics may cycle on graphs containing a cycle even under local additively separable preferences, whereas convergence is guaranteed on trees under LAS preferences, on paths under monotone preferences, and on stars under individually rational preferences [2408.11488].

These subjects study coalition formation **on** graphs, rather than coalition graphs **derived from** partitions or orientations. The terminological overlap is substantial, but the mathematical objects and primary questions are different: auxiliary encoding and recognition in the former, feasibility, optimization, and dynamics in the latter.

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