Middle Graph: Theory and Applications

• Middle graph is a transformation of a finite simple graph G into M(G) with vertex set V(G) ∪ E(G), where adjacency is defined by incidence or edge adjacency.
• It combines subdivision structure with line graph features, enabling precise studies of invariants such as domination numbers, metric dimensions, and coloring parameters.
• Applications include analyzing domination, radio labeling, pebbling, and distinguishing colorings that directly link the structural properties of the base graph with its transformed counterpart.

Searching arXiv for papers on middle graphs and related graph parameters. The middle graph $M(G)$ of a finite simple graph $G$ is the graph with vertex set $V(G)\cup E(G)$ in which adjacency records either incidence between a vertex and an edge of $G$, or adjacency between two edges of $G$. Equivalently, one subdivides each edge of $G$ exactly once and then joins the new subdivision vertices corresponding to adjacent edges. Introduced by Hamada and Yoshimura, this construction is a standard graph transformation that combines subdivision structure with line-graph structure, and it has been studied through domination, metric, coloring, labeling, and pebbling invariants (Banerjee et al., 2024, Kim, 2021, Ghalavand et al., 2022).

1. Definition, notation, and basic examples

For a graph $G=(V(G),E(G))$, the middle graph is defined by

$V(M(G))=V(G)\cup E(G).$

Two vertices $x,y\in V(M(G))$ are adjacent in $M(G)$ if and only if one of the following holds: $G$0 and the corresponding edges are adjacent in $G$1, or $G$2, $G$3, and $G$4 and $G$5 are incident in $G$6 (Kim, 2021, Kazemnejad et al., 2020).

An equivalent constructive description is often more useful. For each edge $G$7, insert a new subdivision vertex $G$8 so that $G$9 is replaced by the path $V(G)\cup E(G)$0. Then, whenever two original edges of $V(G)\cup E(G)$1 share an endpoint, join the corresponding subdivision vertices. In the formal definition, the new vertex $V(G)\cup E(G)$2 is represented simply by the edge $V(G)\cup E(G)$3 itself, regarded as an element of $V(G)\cup E(G)$4 inside $V(G)\cup E(G)$5 (Kim, 2021).

If $V(G)\cup E(G)$6 has order $V(G)\cup E(G)$7 and size $V(G)\cup E(G)$8, then $V(G)\cup E(G)$9 has order $G$0, and its size is

$G$1

where $G$2 is the line graph of $G$3 (Kazemnejad et al., 2020). The original vertex set $G$4 is always an independent set in $G$5, whereas the edge-vertices induce $G$6 (Golpek et al., 24 Jun 2025).

For the path $G$7 with vertices $G$8 and edges $G$9, the middle graph has vertex set $G$0. Each $G$1 is adjacent to the edge-vertices corresponding to incident edges, and consecutive edge-vertices $G$2 are adjacent. Thus $G$3 is a zig-zag graph on $G$4 vertices (Kim, 2021).

For the cycle $G$5, the edge-vertices again form a cycle, and each original vertex is adjacent to the two incident edge-vertices (Kim, 2021, Kazemnejad et al., 2023). For the star $G$6, the edge-vertices induce a clique of size $G$7, because every pair of original edges meets at the center (Kim, 2021).

2. Structural relations with other graph transformations

The middle graph sits naturally between several classical graph constructions. If $G$8 denotes the subdivision graph, then $G$9 is obtained from $G$0 by adding edges between subdivision vertices corresponding to incident edges of $G$1. Consequently, $G$2 is a spanning subgraph of $G$3, while $G$4 appears as a proper induced subgraph on the edge-vertices (Ghalavand et al., 2022).

This viewpoint can be sharpened. If $G$5 is the endline graph obtained from $G$6 by attaching a new pendant edge $G$7 to each original vertex $G$8, then

$G$9

This identity converts questions about vertex colorings of middle graphs into questions about edge colorings of endline graphs and is central in the study of distinguishing colorings (Banerjee et al., 2024).

The total graph $G=(V(G),E(G))$0 is obtained from $G=(V(G),E(G))$1 by adding back the original edges of $G=(V(G),E(G))$2. Thus $G=(V(G),E(G))$3 simultaneously contains $G=(V(G),E(G))$4, $G=(V(G),E(G))$5, and $G=(V(G),E(G))$6 in a single framework, whereas $G=(V(G),E(G))$7 isolates the incidence and edge-adjacency structure without retaining original vertex-vertex adjacency (Ghalavand et al., 2022).

A plausible implication is that middle graphs are especially effective when a problem depends on both vertices and edges of $G=(V(G),E(G))$8, but not on the original edge relation among vertices. This is precisely the regime in which domination, metric, and symmetry parameters on $G=(V(G),E(G))$9 differ qualitatively from their analogues on $V(M(G))=V(G)\cup E(G).$0, $V(M(G))=V(G)\cup E(G).$1, or $V(M(G))=V(G)\cup E(G).$2.

3. Metric, labeling, and symmetry parameters

A basic metric relation is that distances in $V(M(G))=V(G)\cup E(G).$3 are controlled by distances in $V(M(G))=V(G)\cup E(G).$4. For vertices $V(M(G))=V(G)\cup E(G).$5,

$V(M(G))=V(G)\cup E(G).$6

and for a vertex $V(M(G))=V(G)\cup E(G).$7 and an edge $V(M(G))=V(G)\cup E(G).$8,

$V(M(G))=V(G)\cup E(G).$9

These formulas imply that every mixed resolving set of $x,y\in V(M(G))$0 is a vertex resolving set of $x,y\in V(M(G))$1, and therefore

$x,y\in V(M(G))$2

If $x,y\in V(M(G))$3 is a tree with $x,y\in V(M(G))$4 leaves, then

$x,y\in V(M(G))$5

(Ghalavand et al., 2022).

Distinguishing colorings behave in a similarly rigid way. For a connected graph $x,y\in V(M(G))$6 of order $x,y\in V(M(G))$7, the distinguishing chromatic number of the middle graph satisfies

$x,y\in V(M(G))$8

unless $x,y\in V(M(G))$9, in which case

$M(G)$0

The ordinary chromatic number satisfies $M(G)$1, so the symmetry-breaking requirement costs one additional color only in those four exceptional cases (Banerjee et al., 2024).

For radio labeling, the middle graph of a path admits an exact formula. If $M(G)$2, then

$M(G)$3

and if $M(G)$4, then

$M(G)$5

(Bantva, 2018). The proof exploits the center structure of $M(G)$6: for even $M(G)$7, the center is a single vertex, while for odd $M(G)$8, the center is a triangle.

Pebbling produces another exact invariant for cyclic examples. For $M(G)$9,

$G$00

Moreover, Graham’s pebbling conjecture holds for

$G$01

whenever $G$02 and $G$03 (Xia et al., 2017). This places middle graphs of even cycles among the nontrivial families for which Cartesian-product pebbling can be verified exactly.

4. Domination, Roman domination, and exact formulas

For ordinary domination, the fundamental structural result is

$G$04

where $G$05 is the edge cover number of $G$06, provided $G$07 has no isolated vertices (Kazemnejad et al., 2020). As a consequence, if $G$08 has order $G$09 and no isolated vertices, then

$G$10

These bounds are sharp: for paths,

$G$11

for stars $G$12 on $G$13 vertices,

$G$14

for cycles, wheels, and complete graphs,

$G$15

for complete bipartite graphs $G$16 with $G$17,

$G$18

and for friendship graphs $G$19,

$G$20

(Kazemnejad et al., 2020).

Roman domination on middle graphs is unusually clean. If $G$21 has order $G$22, then

$G$23

In the middle-graph formulation on $G$24, a perfect Roman domination number agrees with the ordinary Roman number exactly when there exists a $G$25-function

$G$26

such that every vertex incident to an edge in $G$27 is adjacent to a vertex in $G$28, and the graph

$G$29

is an empty graph (Kim, 2021). For paths,

$G$30

while for cycles,

$G$31

Thus Roman and perfect Roman domination coincide for middle graphs of paths, and for middle graphs of cycles precisely when the cycle length is divisible by $G$32 (Kim, 2021).

These results illustrate a recurring pattern: many domination parameters on $G$33 are governed not by adjacency among original vertices, but by how edge-vertices encode the local edge-incidence geometry of $G$34.

5. Connected, outer-connected, and partial domination

Connected domination in middle graphs is rigid. If $G$35 is connected of order $G$36, then

$G$37

By contrast, the outer-connected domination number satisfies

$G$38

Trees attain the upper bound,

$G$39

and, for connected graphs of order $G$40, the equality $G$41 characterizes trees. Cycles satisfy

$G$42

complete graphs satisfy

$G$43

wheels satisfy

$G$44

and complete bipartite graphs $G$45 with $G$46 satisfy

$G$47

(Kazemnejad et al., 2022).

For total outer-connected domination, if $G$48 is connected of order $G$49 and size $G$50, then

$G$51

Special families again admit exact formulas: for cycles,

$G$52

for star graphs $G$53 of order $G$54,

$G$55

and for paths $G$56 with $G$57,

$G$58

(Kazemnejad et al., 2023).

Partial domination admits an especially sharp characterization. The isolation number of the middle graph satisfies

$G$59

where $G$60 is the size of a smallest maximal matching of $G$61. Consequently,

$G$62

and for complete bipartite graphs,

$G$63

(Zhang et al., 6 Jan 2025). This equivalence with smallest maximal matchings shows that partial domination of middle graphs is controlled directly by edge structure in the original graph.

Another recent direction is paired disjunctive domination. The paired disjunctive domination number of middle graphs has exact values for paths, cycles, wheels, complete graphs, complete bipartite graphs, stars, friendship graphs, double stars, and joins; for example,

$G$64

and

$G$65

for all graphs $G$66 (Golpek et al., 24 Jun 2025).

6. Terminological distinctions and adjacent constructions

The Hamada–Yoshimura middle graph should be distinguished from the middle layer or middle levels graph of the hypercube. For odd dimension $G$67, the middle layer graph is

$G$68

the induced subgraph of the $G$69-cube on all bitstrings of weight $G$70 or $G$71, with edges joining strings at Hamming distance $G$72 (Mütze, 2014, Mütze et al., 2011). Its Hamiltonicity was the content of the middle levels conjecture, proved for all $G$73 in (Mütze, 2014).

A related but different family is the middle cube graph $G$74, defined as the subgraph of $G$75 induced by the layers of weights $G$76 and $G$77. These graphs are the bipartite doubles of odd graphs, are distance-regular, and have fully determined spectra (Dalfó et al., 2016).

This suggests that the phrase “middle graph” is context-sensitive across graph theory and combinatorics on the cube. In the domination, metric, coloring, and labeling literature surveyed above, however, the object is the graph transformation $G$78 with vertex set $G$79. Within that framework, middle graphs form a remarkably stable laboratory: some parameters collapse to simple expressions such as $G$80 or $G$81, while others encode subtler invariants of the base graph, such as edge covers, maximal matchings, leaf structure, or maximum degree (Kim, 2021, Kazemnejad et al., 2020, Kazemnejad et al., 2022).