Coalition Graph: Theory and Applications

• Coalition Graph is defined as either the auxiliary graph of domination partitions or a graph admitting acyclic orientations that enforce coordinated edge directions in induced P4 paths.
• It captures domination synergy by representing coalition partners as partition classes and quantifying invariants like the coalition number and coalition count.
• Variants such as paired, connected, secure, and edge coalition graphs extend the basic concept using alternative domination notions to address diverse graph properties.

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 (Shetty et al., 26 Nov 2025, Le et al., 2015).

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 $\pi=\{V_1,\dots,V_k\}$0 if and only if $\pi=\{V_1,\dots,V_k\}$1 and $\pi=\{V_1,\dots,V_k\}$2 are coalition partners in $\pi=\{V_1,\dots,V_k\}$3 (Shetty et al., 26 Nov 2025).

This construction separates two levels of structure. The underlying graph $\pi=\{V_1,\dots,V_k\}$4 carries domination data, while $\pi=\{V_1,\dots,V_k\}$5 records how the partition classes cooperate to achieve domination. The same graph $\pi=\{V_1,\dots,V_k\}$6 may admit multiple $\pi=\{V_1,\dots,V_k\}$7-partitions, so the associated coalition graph is generally not unique. Haynes et al. also defined the coalition count $\pi=\{V_1,\dots,V_k\}$8 as the maximum number of different coalitions in any $\pi=\{V_1,\dots,V_k\}$9-partition, which measures the density of coalition relations rather than the number of partition blocks (Shetty et al., 26 Nov 2025).

A special case is the singleton coalition graph. If the singleton partition $V_i,V_j$0 is itself a coalition partition, then $V_i,V_j$1 is a singleton-partition graph, and $V_i,V_j$2 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 (Bakhshesh et al., 2023).

2. Realization theory and explicit constructions

A central structural result is a universality theorem: for every graph $V_i,V_j$3, there exist a graph $V_i,V_j$4 and a $V_i,V_j$5-partition $V_i,V_j$6 such that $V_i,V_j$7. Swathi Shetty, Sayinath Udupa N. V., and B. R. Rakshith strengthened this by constructing a graph $V_i,V_j$8 of smaller order and size together with a $V_i,V_j$9-partition $V_i\cup V_j$0 such that $V_i\cup V_j$1 (Shetty et al., 26 Nov 2025).

The construction starts from a clique on the non-isolated vertices of $V_i\cup V_j$2, 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 $V_i\cup V_j$3 are dominating sets by themselves, and the coalition graph $V_i\cup V_j$4 encodes $V_i\cup V_j$5 exactly (Shetty et al., 26 Nov 2025).

Input graph $V_i\cup V_j$6	$V_i\cup V_j$7	$V_i\cup V_j$8
$V_i\cup V_j$9, $c$0 even	$c$1	$c$2
$c$3, $c$4 odd	$c$5	$c$6
$c$7	$c$8	see paper

The singleton setting yields further realization phenomena. For example, the singleton coalition graph chain beginning with $c$9 is

$C(G)$0

while for $C(G)$1 the chain alternates between $C(G)$2 and $C(G)$3. This shows that repeated passage to a singleton coalition graph may stabilize, cycle, or collapse to sparse limiting forms, depending on the starting graph (Bakhshesh et al., 2023).

3. Coalition number, coalition count, and extremal behavior

The invariants $C(G)$4 and $C(G)$5 capture different aspects of coalition structure and are not comparable in general. The complete graph satisfies $C(G)$6 and $C(G)$7, whereas for the 4-cycle one has $C(G)$8 and $C(G)$9. Thus maximal partition size need not imply many coalition edges, and a partition with relatively few blocks can still induce a dense coalition graph (Shetty et al., 26 Nov 2025).

For graphs with no isolated vertices and $c$0 full vertices, the coalition count satisfies

$c$1

where $c$2 is the domatic number; this bound is sharp for stars and complete graphs. A full characterization is available for the smallest positive value: $c$3 where $c$4 is the independence number. Equivalently, all such graphs are isomorphic to $c$5 for appropriate $c$6 (Shetty et al., 26 Nov 2025).

Coalition graphs arising from special partition types can also be constrained quite rigidly. If a graph has exactly one full vertex $c$7, minimum degree $c$8, and coalition number $c$9, then $\pi$0, and the coalition graph of a suitable $\pi$1-partition is

$\pi$2

For singleton-partition graphs with no full vertices, the bound $\pi$3 further links coalition density to independence structure (Shetty et al., 26 Nov 2025).

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 $\pi$4 or identifying all $\pi$5 with a prescribed $\pi$6—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	$\pi$7-partition	$\pi$8
Connected domination	connected coalition partition	$\pi$9
Secure domination	$CG(G,\pi)$0-partition	$CG(G,\pi)$1
Edge domination	$CG(G,\pi)$2-partition of $CG(G,\pi)$3	$CG(G,\pi)$4

For paired coalitions, $CG(G,\pi)$5 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 $CG(G,\pi)$6 and $CG(G,\pi)$7. For cycles, the possible paired coalition graphs are $CG(G,\pi)$8, $CG(G,\pi)$9, $\{V_1,\dots,V_k\}$0, $\{V_1,\dots,V_k\}$1, and $\{V_1,\dots,V_k\}$2. For trees with $\{V_1,\dots,V_k\}$3, the paired coalition graph is always a star $\{V_1,\dots,V_k\}$4 (Samadzadeh et al., 2024).

For connected coalitions, the graph $\{V_1,\dots,V_k\}$5 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 $\{V_1,\dots,V_k\}$6 for $\{V_1,\dots,V_k\}$7 together with 22 finite graphs of order at most 6, including $\{V_1,\dots,V_k\}$8, $\{V_1,\dots,V_k\}$9, $\pi=\{V_1,\dots,V_k\}$00, $\pi=\{V_1,\dots,V_k\}$01, and $\pi=\{V_1,\dots,V_k\}$02. In the earlier foundational study of connected coalitions, all trees were shown to satisfy $\pi=\{V_1,\dots,V_k\}$03, and polynomial-time algorithms were given to determine whether $\pi=\{V_1,\dots,V_k\}$04 or $\pi=\{V_1,\dots,V_k\}$05 (Dobrynin et al., 4 Sep 2025, Alikhani et al., 2023).

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 $\pi=\{V_1,\dots,V_k\}$06, there exist $\pi=\{V_1,\dots,V_k\}$07 and $\pi=\{V_1,\dots,V_k\}$08 with $\pi=\{V_1,\dots,V_k\}$09. This is a secure-domination analogue of the universality phenomenon for ordinary coalition graphs (Shetty et al., 26 Nov 2025).

For edge coalitions, the partition is over $\pi=\{V_1,\dots,V_k\}$10 rather than $\pi=\{V_1,\dots,V_k\}$11. The edge coalition graph $\pi=\{V_1,\dots,V_k\}$12 has partition classes of edges as vertices. In this setting, $\pi=\{V_1,\dots,V_k\}$13 if and only if $\pi=\{V_1,\dots,V_k\}$14, $\pi=\{V_1,\dots,V_k\}$15 if and only if $\pi=\{V_1,\dots,V_k\}$16, and $\pi=\{V_1,\dots,V_k\}$17 if and only if $\pi=\{V_1,\dots,V_k\}$18. For stars $\pi=\{V_1,\dots,V_k\}$19, the only $\pi=\{V_1,\dots,V_k\}$20-partition is the singleton partition, and the edge coalition graph is $\pi=\{V_1,\dots,V_k\}$21 (Besharati et al., 26 Jul 2025).

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 $\pi=\{V_1,\dots,V_k\}$22, and every $\pi=\{V_1,\dots,V_k\}$23-partition is a coalition partition, so $\pi=\{V_1,\dots,V_k\}$24. However, the supplied data do not define an associated “perfect coalition graph”; current results emphasize partition existence, extremal values, and structural characterizations instead (Mojdeh et al., 2024).

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 $\pi=\{V_1,\dots,V_k\}$25, the end-edges $\pi=\{V_1,\dots,V_k\}$26 and $\pi=\{V_1,\dots,V_k\}$27 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 $\pi=\{V_1,\dots,V_k\}$28 on its vertex set such that every induced $\pi=\{V_1,\dots,V_k\}$29 satisfies $\pi=\{V_1,\dots,V_k\}$30 if and only if $\pi=\{V_1,\dots,V_k\}$31. In some literature these are called one-in-one-out graphs (Le et al., 2015).

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 $\pi=\{V_1,\dots,V_k\}$32 to be oriented in opposition rather than in the same direction (Le et al., 2015).

Recognition is open in general, but several restricted classes admit complete characterizations. For a graph $\pi=\{V_1,\dots,V_k\}$33, the auxiliary graph $\pi=\{V_1,\dots,V_k\}$34 has as vertices the ordered pairs $\pi=\{V_1,\dots,V_k\}$35 where $\pi=\{V_1,\dots,V_k\}$36 is an end-edge of some induced $\pi=\{V_1,\dots,V_k\}$37; two such vertices are adjacent when one is the reverse of the other, or when they appear as the end-edges of an induced $\pi=\{V_1,\dots,V_k\}$38. If $\pi=\{V_1,\dots,V_k\}$39 is $\pi=\{V_1,\dots,V_k\}$40-free, then

$\pi=\{V_1,\dots,V_k\}$41

yielding an $\pi=\{V_1,\dots,V_k\}$42-time recognition algorithm. For distance-hereditary graphs, the following are equivalent: $\pi=\{V_1,\dots,V_k\}$43 is a coalition graph, $\pi=\{V_1,\dots,V_k\}$44 is bipartite, $\pi=\{V_1,\dots,V_k\}$45 is $\pi=\{V_1,\dots,V_k\}$46-free, and $\pi=\{V_1,\dots,V_k\}$47 is a comparability graph; recognition and orientation are then possible in $\pi=\{V_1,\dots,V_k\}$48 time (Le et al., 2015).

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 (Le et al., 2015).

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 $\pi=\{V_1,\dots,V_k\}$49 and a valuation $\pi=\{V_1,\dots,V_k\}$50, and seeks a partition of $\pi=\{V_1,\dots,V_k\}$51 into connected subsets maximizing $\pi=\{V_1,\dots,V_k\}$52. 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 $\pi=\{V_1,\dots,V_k\}$53- and $\pi=\{V_1,\dots,V_k\}$54-minor-free graphs (Voice et al., 2011).

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 $\pi=\{V_1,\dots,V_k\}$55-approximation for $\pi=\{V_1,\dots,V_k\}$56-minor-free graphs, and a $\pi=\{V_1,\dots,V_k\}$57-approximation for bounded-degree graphs (Bachrach et al., 2011).

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 $\pi=\{V_1,\dots,V_k\}$58 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 (Bistaffa et al., 2016).

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 (Fanelli et al., 2024).

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.