The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$ is the graph whose vertices correspond to the $i(G)$-sets, and where two $i(G)$-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are $i$-graph realizable, that is, given a target graph $H$, there does not necessarily exist a source graph $G$ such that $H$ is the $i$-graph of $G$. We consider a class of graphs called"theta graphs": a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are $i$-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are $i$-graphs, and show that all $3$-connected cubic bipartite planar graphs are $i$-graphs.


Abstract
The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set.The i-graph of G, denoted I (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)sets are adjacent if and only if they differ by two adjacent vertices.Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H ∼ = I (G).We consider a class of graphs called "theta graphs": a theta graph is the union of three internally disjoint nontrivial

Introduction
Our main topic is the characterization of theta graphs that are obtainable as reconfiguration graphs with respect to the minimum independent dominating sets of some graph.
In graph theory, reconfiguration problems are often concerned with solutions to a specific problem that are vertex subsets of a graph.When this is the case, the reconfiguration problem can be viewed as a token manipulation problem, where a solution subset is represented by placing a token at each vertex of the subset.Each solution is represented as a vertex of a new graph, referred to as a reconfiguration graph, where adjacency between vertices corresponds to a predefined token manipulation rule called the reconfiguration step.The reconfiguration step we consider here consists of sliding a single token along an edge between adjacent vertices belonging to different solutions.
More formally, given a graph G, the slide graph of G is the graph H such that each vertex of H represents a solution of some problem on G, and two vertices u and v of H are adjacent if and only if the solution in G corresponding to u can be transformed into the solution corresponding to v by sliding a single token along the edge uv ∈ E(G).See [7] for a survey on reconfiguration of colourings and dominating sets in graphs.
We use the standard notation of α(G) for the independence number of a graph G.The independent domination number i(G) of G is the minimum cardinality of a maximal independent set of G, or, equivalently, the minimum cardinality of an independent domination set of G.An independent dominating set of G of cardinality i(G) is also called an i-set of G, or an i(G)-set.An α-set of G, or an α(G)-set, is defined similarly.When i(G) = α(G), we say that G is well-covered.
Given a graph G, we consider the slide graph I (G) of G, formally defined in Section 2 below, with respect to its i-sets.Our main result, Theorem 5.3, concerns the class Θ of "theta graphs" for which we characterize, in Sections 5 and 6, those graphs H for which there exists a graph G ∈ Θ such that H ∼ = I (G).We state known results required here in Section 3. We introduce the technique we use for theta graphs in Section 4, where we apply it to the simpler problem of characterizing line graphs and claw-free graphs that are realizable as i-graphs.In Section 7.1 we exhibit a graph that is neither a theta graph nor an i-graph, and in Section 7.2 we show that certain planar graphs are i-graphs and α-graphs.We conclude with some open problems in Section 8.
In general, we follow the notation of [3].For other domination principles and terminology, see [4,5].

i-Graphs and Theta Graphs
The i-graph of a graph G, denoted I (G) = (V (I (G)), E(I (G))), is the graph with vertices representing the minimum independent dominating sets of G (that is, the i-sets of G), and where u, v ∈ V (I (G)), corresponding to the i(G)-sets S u and S v , respectively, are adjacent in I (G) if and only if there exists xy ∈ E(G) such that S u = (S v − x) ∪ {y}.Imagine that there is a token on each vertex of an i-set S of G. Then S is adjacent, in I (G), to an i(G)-set S ′ if and only if a single token can slide along an edge of G to transform S into S ′ .Similarly, the α-graph A (G) of a graph G is the slide reconfiguration graph with vertices representing the α(G)-sets, and where adjacency is defined as for the i-graph.
In Section 5 we present several constructions for i-graphs that are also constructions for α-graphs.
We say H is an i-graph, or is i-graph realizable, if there exists some graph G such that I (G) ∼ = H.Moreover, we refer to G as the seed graph of the i-graph H. Going forward, we mildly abuse notation to denote both the i-set X of G and its corresponding vertex in H as X, so that X ⊆ V (G) and X ∈ V (H).
In acknowledgment of the slide-action in i-graphs, given i-sets X = {x 1 , x 2 , . . ., x k } and Y = {y 1 , x 2 , . . .x k } of G with x 1 y 1 ∈ E(G), we denote the adjacency of X and Y in I (G) as X x 1 y 1 ∼ Y , where we imagine transforming the i-set X into Y by sliding the token at x 1 along an edge to y 1 .When discussing several graphs, we use the notation X x 1 y 1 ∼ G Y to specify that the relationship is on G.More generally, we use x ∼ y to denote the adjacency of vertices x and y (and x ̸ ∼ y to denote non-adjacency); this is used in the context of both the seed graph and the target graph.
The study of i-graphs was initiated by Teshima in [9].In [2], Brewster, Mynhardt and Teshima investigated i-graph realizability and proved some results concerning the adjacency of vertices in an i-graph and the structure of their associated i-sets in the seed graph.They presented the three smallest graphs that are not i-graphs: the diamond graph D = K 4 − e, K 2,3 and the graph κ, which is K 2,3 with an edge subdivided.They showed that several graph classes, like trees and cycles, are i-graphs.They demonstrated that known i-graphs can be used to construct new i-graphs and applied these results to build other classes of i-graphs, such as block graphs, hypercubes, forests, cacti, and unicyclic graphs.

Previous Results
To begin, we state several useful observations and lemmas from [2,9] about the structure of i-sets within given i-graphs.Given a set S ⊆ V (G) and a vertex v ∈ S, the private neighbourhood of v with respect to S is the set pn(v, S) = N [v] − N [S − {v}], and the external private neighbourhood of v with respect to S is the set epn(v, S) = pn(v, S) − {v}.Observation 3.1 [2,9] Let G be a graph and For some path X 1 , X 2 , . . ., X k in H, at most one vertex of the i-set is changed at each step, and so X 1 and X k differ on at most k − 1 vertices.This is yields the following immediate observation.Observation 3.2 [2,9] Let G be a graph and Combining the results from Lemma 3.3 with Observation 3.2 yields the following observation for vertices of i-graphs at distance 2.
Observation 3.4 [2,9] Let G be a graph and Lemma 3.5 [2,9] Let G be a graph and Proposition 3.6 [2,9] Let G be a graph and H = I (G).Suppose H has an induced C 4 with vertices X, A, B, Y , where XY, AB / ∈ E(H).Then, without loss of generality, the set composition of X, A, B, Y in G, and the edge labelling of the induced C 4 in H, are as in Figure 1.
Reconfiguration structure of an induced C 4 subgraph from Proposition 3.6.
We state the above-mentioned results regarding D, K 2,3 , and κ here for referencing.
On the other hand, the house graph H = Θ ⟨1, 2, 3⟩ in Figure 2  The next result shows that maximal cliques in i-graphs can be replaced by arbitrarily larger maximal cliques to form larger i-graphs.Lemma 3.9 (Max Clique Replacement Lemma) [2,9] Let H be an i-graph with a maximal m-vertex clique, K m .The graph H w formed by adding a new vertex w * adjacent to all of K m is also an i-graph.
The Deletion Lemma below shows that the class of i-graphs is closed under vertex deletion, that is, every induced subgraph of an i-graph is also an i-graph.Lemma 3.10 (Deletion Lemma) [2,9] If H is a nontrivial i-graph, then any induced subgraph of H is also an i-graph.Corollary 3.11 [2,9] If H is not an i-graph, then any graph containing an induced copy of H is also not an i-graph.
When visualizing the connections between the i-sets of a graph G, it is sometimes advantageous to consider its complement G instead.From a human perspective, it is curiously easier see to which vertices a vertex v is adjacent, rather than to which vertices v is nonadjacent.This is especially true when i(G) = 2 or 3, when we may interpret the adjacency of i-sets of G as the adjacencies of edges and triangles (i.e.K 3 ), respectively, in G.In the following sections we examine how the use of graph complements can be exploited to construct the i-graph seeds for certain classes of line graphs, theta graphs, and maximal planar graphs.
4 Line Graphs and Claw-free Graphs Consider a graph G with i(G) = 2 and where X = {u, v} is an i-set of G.In G, u and v are adjacent, so X is represented as the edge uv.Moreover, no other vertex w is adjacent to both vertices of X in G; otherwise, {u, v, w} is independent in G, contrary to X being an i-set.Now consider the line graph L(G) of G.If X = {u, v} is an i-set of G, then e = uv is an edge of G and hence e is a vertex of L(G).Thus, the i-sets of G correspond to a subset of the vertices of L(G).In the case where G is triangle-free (that is, G has no independent sets of cardinality 3), these i-sets of G are exactly the vertices of L(G).Now suppose Y is an i-set of G adjacent to X; say, X uw ∼ Y , so that Y = {v, w}.Then, in G, f = vw is an edge, and so in L(G), f ∈ V (L(G)).Since e and f are both incident with v in G, ef ∈ E(L(G)).That is, for i-sets X and Y of a well-covered graph G with i(G) = α(G) = 2, X ∼ Y if and only if X and Y correspond to adjacent vertices in L(G).Thus I (G) ∼ = L(G).
In the example illustrated in Figure 3  The connection between graphs with i(G) = 2 and line graphs helps us not only understand the structure of I (G), but also lends itself towards some interesting realizability results.We follow this thread for the remainder of this section, and build towards determining the i-graph realizability of line graphs and claw-free graphs.The straightforward proof of the following lemma can be found in [9] and is omitted here.Proof.Suppose H is an i-graph.By Proposition 3.7 and Corollary 3.11, H is D-free.
Conversely, suppose that H is D-free.If H is complete, then H is the i-graph of itself.So, assume that H is not complete.Say H is the line graph of some graph F , where we may assume F has no isolated vertices (as isolated vertices do not affect line graphs).Since H is D-free and connected, F has no triangles by Lemma 4.1.Since F has edges (which it does since H exists), α(F ) ≤ 2.Moreover, as F is connected, F has no universal vertices, and so i(F ) ≥ 2. Thus, i(F ) = α(F ) and F is well-covered.It follows that every edge of F corresponds to an i-set of F .Since H is the line graph of F , it is the i-graph of F .
Finally, if we examine Beineke's forbidden subgraph characterization of line graphs (see [1] or [3,Theorem 6.26]), we note that eight of the nine minimal non-line graphs contain an induced D and are therefore not i-graphs.The ninth minimal non-line graph is the claw, K 1,3 .Thus, D-free claw-free graphs are D-free line graphs, hence i-graphs.
Corollary 4.3 Let H be a connected claw-free graph.Then H is an i-graph if and only if H is D-free.
While Theorem 4.2 and Corollary 4.3 reveal the i-graph realizability of many famous graph families (including another construction for cycles, which are connected, claw-free, and D-free), the realizability problem for graphs containing claws remains unresolved.Moreover, among clawed graphs are the theta graphs which we first alluded to in Section 2 as containing three of the small known non-i-graphs.In the next section we apply similar techniques with graph complements to construct all theta graphs that are i-graphs.

Theta Graphs From Graph Complements
are the triangles of G, for notational simplicity we refer to X and Y as triangles.In G, the triangle X can be transformed into the triangle Y by removing the vertex u and adding in the vertex v (where u ̸ ∼ G v). Thus, we say that two triangles are adjacent if they share exactly one edge.Moreover, since two i-sets of a graph G with i(G) = 3 are adjacent if and only if their associated triangles in G are adjacent, we use the same notation for i-set adjacency in G as triangle adjacency in G; that is, the notation X ∼ Y represents both i-sets X and Y of G being adjacent, and triangles X and Y of G being adjacent.
In the following sections we use triangle adjacency to construct complement seed graphs for the i-graphs of theta graphs; that is, a graph G such that I (G) is isomorphic to some desired theta graph.Before proceeding with these constructions, we note some observations which will help us with this process.
and only if G is nonempty and has an edge that does not lie on a triangle.
If G has an edge uv that does not lie on a triangle, then {u, v} is independent and dominating in G, and so i(G) ≤ 2. When building our seed graphs G with i(G) = 3, it is therefore necessary to ensure that every edge of the complement G belongs to a triangle.
Suppose that S = {u, v, w, x} is such a clique of G.Then, for example, x is undominated by {u, v, w} in G, and so {u, v, w} is not an i-set of G. Conversely, suppose that {u, v, w} is a triangle in a graph G with i(G) = 3.By attaching a new vertex x to all of {u, v, w} in G, we remove {u, v, w} as an i-set of G, while keeping all other i-sets of G.This observation proves to be very useful in the constructions in the following section: we now have a technique to eliminate any unwanted triangles in G (and hence i-sets of G) that may arise.Note that this technique is an application of the Deletion Lemma (Lemma 3.10) in G instead of in G.
The use of triangle adjacency in a graph G to determine i-set adjacency in G provides a key technique to resolve the question of which theta graphs are i-graphs.In our main result, Theorem 5.3, we show that all theta graphs except the seven listed exceptions are i-graphs.Using this method of complement triangles, the proofs of the lemmas for the affirmative cases make up most of the remainder of Section 5.The proofs of the lemmas for the seven negative cases are given in Section 6.
Theorem 5.3 A theta graph is an i-graph if and only if it is not one of the seven exceptions listed below: Table 1 summarizes the cases used to establish Theorem 5.3 and their associated results.
We can further exploit previous results to see that all graphs Θ ⟨1, 2, ℓ⟩ for ℓ ≥ 3 are igraphs by taking a cycle C n with n ≥ 4, and replacing one of its maximal cliques (i.e. an edge) with a K 3 .By the Max Clique Replacement Lemma (Lemma 3.9), the resultant Θ ⟨1, 2, n − 1⟩ is also an i-graph.For reference, we explicitly state this result as a lemma.

Construction of
In Figure 5 below, we provide a first example of the technique we employ repeatedly throughout this section to construct our theta graphs.To the left is a graph G, where each of its nine triangles corresponds to an i-set of its complement G.The resultant i-graph of G, I (G) = Θ ⟨1, 4, 5⟩, is presented on the right.For consistency, we use X and Y to denote the triangles corresponding to the degree 3 vertices in the theta graphs in this example, as well as all constructions to follow.We proceed now to the general construction of a graph G with As it is our first construction using this triangle technique, we provide the construction and proof for Lemma 5.6 with an abundance of detail.Proof.To begin, notice that since in G, the vertices W = {w 0 , w 1 , . . ., w k , w k+1 } form a wheel on at least five vertices, H ≇ K 4 .Likewise, the graph induced by {w 0 , w 1 , w 2 , w 3 , v 1 , v 2 , . . ., v ℓ−2 } in G is also a wheel on ℓ + 2 vertices, where w 2 is the central hub, and so it too contains no K 4 .Therefore, G is K 4 -free and the triangles of G are precisely the maximal cliques of G, and so ω(G) = i(G) = α(G) = 3.Since the i-sets of G are identical to its α-sets, I (G) = A (G), and so for ease of notation, we will refer only to I (G) throughout the remainder of this proof.
We label the triangles as in Figure 6 by dividing them into two collections.The first are the triangles composed only of the vertices from W and each containing w 0 : let The remainder are the triangles with vertex sets not fully contained in H: We now show that the required adjacencies hold.From the construction of G, the following are immediate for I (G): Hence, we need only show that there are no additional unwanted edges generated in the construction of I (G).
Since G is a planar graph and all of its triangles are facial (that is, the edges of the K 3 form a face in the plane embedding in Figure 6), each triangle is adjacent to at most three others.From (i) -(iii) above, triangles X and Y are both adjacent to the maximum three (and hence deg Recall that to be adjacent, two triangles share exactly two vertices.Notice that the triangles of A are composed entirely of vertices from W − {w 2 }, and that for 2 Therefore no triangle of B is adjacent to any triangle of A. It is similarly easy to see that there are no additional unwanted adjacencies between two triangles of A or two triangles of B.
We conclude that the graph G generated by Construction 5.5 yields Before we proceed with the remainder of the theta graph constructions, let us return to Figure 6 to notice the prominence of the wheel subgraph in the complement seed graph G.In the constructions throughout this paper, this wheel subgraph will appear repeatedly; indeed, all of the complement seed graphs for the i-graphs of theta graphs have a similar basic form: begin with a wheel, add a path of some length, and then add some collection of edges between them.As stated without proof in Lemma 5.7, Figure 7 below demonstrates that a wheel in a triangle-based complement seed graph G corresponds to a cycle in the igraph of G. Using this result, in our later constructions with a wheel subgraph, we already have two of the three paths of a theta graph formed.Hence, we need only confirm that whatever unique additions are present in a given construction form the third path in the i-graph.
We note the following analogous -although less frequently applied -result for fans of the form K 1 + P k , as illustrated in Figure 8.
We label the resultant (planar) graph G.
The graph G from Construction 5.9 such that I (G) = Θ ⟨2, 2, ℓ⟩ for ℓ ≥ 6.As with our other constructions, the triangles of G are its smallest maximal cliques, and so i(G) = 3.However, we now employ a technique of adding vertices to create K 4 's in G and eliminate any "unwanted" triangles that might arise in our construction.In (b), the addition of z 1 and z 2 eliminate triangles {w 1 , w 4 , v 1 } and {w 2 , w 3 , v 2 }, respectively.Similarly, in (a), z prevents {w 1 , w 4 , v ℓ−3 } from being a maximal clique of G and hence an i-set of G.The unfortunate trade-off in this triangle-elimination technique is that the remaining triangles are no longer α-sets; the constructions work only for i-graphs, not α-graphs.
Proof.We only prove the lemma for case where ℓ ≥ 6; the single case where ℓ = 5 is adequately illustrated in Figure 10 and details can be found in [9].
As in the previous constructions, since each edge of G belongs to a triangle and some triangles are not contained in K 4 's, these triangles of G form the smallest maximal cliques of G.We label these triangles as in Figure 9; in particular, It is straightforward to verify that these ℓ + 3 sets are precisely the maximal cliques of G of order 3 and, hence, the i-sets of G. Therefore they form the vertex set of I (G).Moving to the edges of I (G), the following adjacencies are clear from Figure 9: As for its vertex set, it is straightforward to verify that the edge set of I (G) consists of precisely the edges listed in (i) − (iii).We conclude that I (G) = Θ ⟨2, 2, ℓ⟩.

Construction of
For many of the results going forward, we apply small modifications to previous constructions.In the first of these, we begin with the graphs from Construction 5.9, which were used to find i-graphs for Θ ⟨2, 2, 5⟩ and Θ ⟨2, 2, ℓ⟩ for ℓ ≥ 6, and expand the central wheel used in there to build i-graphs for Θ ⟨2, 3, 5⟩ and Θ ⟨2, 3, ℓ⟩ for ℓ ≥ 6.(a) If ℓ ≥ 6, begin with a copy of the graph G from Construction 5.9.Subdivide the edge w 1 w 4 , adding the new vertex w 5 .Join w 5 to w 0 , so that w 0 , w 1 , . . ., w 5 forms a wheel.Delete the vertex z.
(b) If ℓ = 5, begin with a copy of the graph G from Construction 5.9.Subdivide the edge w 1 w 4 , adding the new vertex w 5 .Join w 5 to w 0 , so that w 0 , w 1 , . . ., w 5 forms a wheel.Delete the vertex z 1 .

Construction of G for Θ ⟨2, 4, 4⟩
In the following construction for a graph G with I (G) = Θ ⟨2, 4, 4⟩, we again apply the technique of adding vertices to eliminate unwanted triangles.Proof.As shown in Figure 13, the triangles of G forming maximal cliques of G, and therefore the i-sets of G, are labelled them as follows: Similarly to the construction for Θ ⟨2, 2, 5⟩ in the proof of Lemma 5.10, the vertices z and z ′ are added to ensure that {v, w 2 , w 3 } and {w 0 , w 1 , w 4 }, respectively, are not maximal cliques in G, and hence are not i-sets of G.
It can be seen that there are no other triangles in G beyond the nine listed above; we omit the details, which can be found in [9,Lemma 5.19].
From Figure 13, the following triangle adjacencies are immediate: It is again straightforward to verify that there are no additional edges in I (G) than those listed above.We conclude that I (G) = Θ ⟨2, 4, 4⟩ as required.
Notice that Construction 5.13 is our first theta graph construction that is not planar as it has a K 3,3 minor.Indeed, with the exception of the constructions that are based upon Construction 5.13, all of our i-graph constructions use planar complement seed graphs.
(ii) Do all i-graphs with largest induced stars of K 1,3 , always have a planar graph complement construction?
A large target graph requires a large seed graph in order to generate a sufficient number of unique i-sets.Can a target graph become too dense to allow for a planar graphcomplement construction?
Moving forward, we will no longer explicitly check that there are no additional unaccounted for triangles in our constructions.Should the construction indeed result in triangles of G that produce extraneous vertices in I (G), we can easily remove them using the Deletion Lemma (Lemma 3.10).(b) Begin with a copy of the graph G 2,4,5 from Construction 5.15 (a).Subdivide the edge w 1 w 6 , adding the new vertex w 7 .Join w 7 to w 0 , so that w 0 , w 1 , . . ., w 7 forms a wheel.Call this graph G 2,5,5 .Construction 5.17 See Figure 15.Begin with a copy of the graph G 2,3,ℓ from Construction 5.11 (a) for ℓ ≥ 5. Subdivide the edge w 1 w 5 k − 3 times (for k ≤ ℓ), adding the new vertices w 6 , w 7 , . . ., w k+2 .Join w 6 , w 7 , . . ., w k+2 to w 0 , so that w 0 , w 1 , . . ., w k+2 forms a wheel.Call this graph G 2,k,ℓ .

Θ ⟨3, k, ℓ⟩
In the rest of Section 5 we only give the constructions and their associated figures, and state the lemmas without proof.Details can be found in [9, Chapter 5].

5.3.1
The graph G for Θ ⟨3, 3, 4⟩ The triangles 16, and the obvious paths formed by them, illustrate the following lemma.Although the general construction still applies for the case when ℓ = 6, we include a separate figure for the construction of Θ ⟨3, 3, 6⟩ for reference below, because of the additional complication that now v 1 = v ℓ−5 , and so the single vertex now has dual roles in the construction.and w 8 to w 0 , so that w 0 , w 1 , . . ., w 8 forms a wheel.Call this graph G 3,5,5 .Notice that subdividing only once in Construction 5.28 (adding only w 7 and not w 8 ) gives an alternative (planar) construction for Θ ⟨3, 4, 5⟩.
Construction 5.30 Refer to Figure 22.Begin with a copy of the graph G 3,4,4 from Construction 5.24.Subdivide the edge u 1 w 3 , adding the new vertex u 2 .Join u 2 to w 0 , so that w 0 , w 1 , w 2 , u 1 , u 2 , w 3 , . . ., w 6 forms a wheel.Call this graph G 4,4,4 .Construction 5.32 Begin with a copy of the graph G 3,3,5 from Construction 5.20.For k = 4 subdivide the edge w 1 w 6 once, adding the vertex w 7 ; for k = 5, subdivide a second time, adding the vertex w 8 .For j = 4, subdivide the edge w 2 w 3 , adding the vertex u 1 ; for j = 5 (j ≤ k), subdivide a second time, adding the vertex u 2 .Connect all new vertices to w 0 to form a wheel.Call this graph G j,k,5 for 4 ≤ j ≤ k ≤ 5.
An example of the construction of G 5,5,5 is given in Figure 23 below.

Construction of
Construction 5.34 Begin with a copy of the graph G 3,3,ℓ from Construction 5.22 for Θ ⟨3, 5, ℓ⟩ for ℓ ≥ 6.For 3 ≤ k ≤ ℓ, subdivide the edge w 1 w 6 k − 3 times, adding the new vertices w 7 , w 8 , . . ., w k+3 .Join each of w 7 , w 8 , . . ., w k+3 to w 0 , so that w 0 , w 1 , . . ., w k+3 forms a wheel.Then, for 3 ≤ j ≤ k, subdivide the edge w 2 w 3 j − 3 times, adding the new vertices u 1 , u 2 , . . ., u j−3 .Again, join each of u 1 , u 2 , . . ., u j−3 to w 0 to form a wheel.Call this graph G j,k,ℓ for 3 ≤ j ≤ k ≤ ℓ and ℓ ≥ 6.The lemmas above imply the sufficiency of Theorem 5.3: if a theta graph is not one of seven exceptions listed, then it is an i-graph.In the next section, we complete the proof by examining the exception cases.
From Proposition 1, the composition of the following i-sets of G are immediate: where k ≥ 3 and y 1 , y 2 , y 3 are three distinct vertices in G − X.These sets are illustrated in blue in Figure 25.This leaves only the composition of C 2 and C 3 (in red) to be determined.As we construct C 2 , notice first that y 3 ∈ C 2 ; otherwise, if say some other z ∈ C 2 so that C 1 and XC 2 ∈ E(H).Thus, a token on one of {x 1 , x 2 , x 4 , . . ., x k } moves in the transition from C 1 to C 2 .We consider three cases.
Case 1: The token on x 1 moves.If C 1 , contradicting the distance requirement between i-sets from Observation 3.2.Moreover, from the composition of B, x 1 ̸ ∼ y 2 , and so C 1 However, since x 3 ∼ y 3 , we have that A Case 2: The token on x 2 moves.An argument similar to Case 1 constructs C 2 = {x 1 , y 2 , y 3 , . . ., x k }, with B Case 3: The token on x i for some i ∈ {4, 5, . . ., k} moves.From the compositions of X and Y , x i is not adjacent to any of {x 3 , y 1 , y 2 }, so the token at x i moves to some other vertex, say z, so that C 1 In all cases, we fail to construct a graph G with I (G) ∼ = Θ ⟨2, 2, 4⟩ and so conclude that no such graph exists.

6.2
Θ ⟨2, 3, 3⟩ is not an i-Graph Proposition 6.2 The graph Θ ⟨2, 3, 3⟩ is not i-graph realizable.Proof.To begin, we proceed similarly to the proof of Proposition 6.1: suppose to the contrary that Θ ⟨2, 3, 3⟩ is realizable as an i-graph, and that H = Θ ⟨2, 3, 3⟩ ∼ = I (G) for some graph G. Label the vertices of H as in Figure 26.As before, the corresponding i-sets in blue are established from previous results, and those in red are yet to be determined.Moreover, from the composition of these four blue i-sets, we observe that for each i ∈ {1, 2, 3}, x i ∼ y j if and only if i = j.
Unlike the construction for Θ ⟨2, 2, 4⟩, we no longer start with knowledge of the exact composition of Y .We proceed with a series of observations on the contents of the various i-sets: and y 3 ∈ C 2 by three applications of Proposition 3.6.
Using similar arguments, we determine that C 2 y 3 y 1 ∼ Y , and that C 1 ∼ Y , we have that Y = {y 1 , z, x 3 , x 4 . . ., x k }.However, from C 2 y 3 y 1 ∼ Y , we also have that Y = {y 1 , x 2 , w, x 4 , . . ., x k }.As we have already established that z ̸ = x 2 and w ̸ = x 3 , we arrive at two contradicting compositions of Y .Thus, no such graph G exists, and we conclude that Θ ⟨2, 3, 3⟩ is not an i-graph.

6.3
Θ ⟨2, 3, 4⟩ is not an i-Graph Proposition 6.3 The graph Θ ⟨2, 3, 4⟩ is not i-graph realizable.Proof.The construction for our contradiction begins similarly to that of Θ ⟨2, 3, 3⟩ in Proposition 6.2.As before, we illustrate the graph in Figure 27, labelling the known sets in blue, and those yet to be determined in red.Given the similarity of Θ ⟨2, 3, 3⟩ and Θ ⟨2, 3, 4⟩, many of the observations from Proposition 6.2 carry through to our current proof.In particular, all of (i) -(iv) hold here, including that y 1 / ∈ C 2 from (ii).Moreover, the compositions of Y and B 2 also hold, where z is some vertex with z ̸ ∈ {y 1 , x 2 , y 2 }.
We now attempt to build C 2 .From Proposition 3.6, since X ̸ ∼ C 2 , y 3 ∈ C 2 (and x 3 ̸ ∈ C 2 ).From the distance requirement of Observation 3.2, |X − C 2 | ≤ 2, and so at least one of x 1 or x 2 is in C 2 .Recall from the construction for Proposition 6.2 that A and so z is adjacent to both x 1 and x 2 .Hence, z ̸ ∈ C 2 .
Gathering these results shows that none of {x 3 , z, y 1 } are in C 2 , and thus, d(C 2 , Y ) ≥ 3, contradicting the distance requirement of Observation 3.2.We conclude that no graph G exists such that I (G) = Θ ⟨2, 3, 4⟩.
Proof.Let H be the theta graph Θ ⟨3, 3, 3⟩, with vertices labelled as in Figure 28.Suppose to the contrary that there exists some graph G such that H is the i-graph of G; that is, ∼ Y , which is not so.We therefore conclude that x 4 ∈ Y , and likewise ∈ Y , we have that y 1 ∈ Y .Similarly, y 2 , y 3 ∈ Y .Thus, Y = {y 1 , y 2 , y 3 , x 3 , . . ., x k }.Moreover, A 2 is obtained from A 1 by replacing one of x 2 or x 3 , by y 2 or y 3 , respectively.Say, A 1 x 2 y 2 ∼ A 2 so that A 2 = {y 1 , y 2 , x 3 , . . ., x k }.Now, however, we have that B 1 This completes the proof of Theorem 5.3.

Other Results
In this section we first display a graph that is neither a theta graph nor an i-graph, and then use the method of graph complements to show that every cubic 3-connected bipartite planar graph is an i-graph.

A Non-Theta Non-i-Graph
So far, every non-i-graph we have observed is either one of the seven theta graphs from Theorem 5.3, or contains one of those seven as an induced subgraph (as per Corollary 3.11).This leads naturally to the question of whether theta graphs provide a forbidden subgraph characterization for i-graphs.Unfortunately, this is not the case.
Consider the graph T in Figure 29: it is not a theta graph, and although it contains several theta graphs as induced subgraphs, none of those induced subgraphs are among the seven non-i-graph theta graphs.In Proposition 7.1 we confirm that T is not an i-graph.Proof.We proceed similarly to the proofs for the theta non-i-graphs in Section 6.Let T be the graph in Figure 29, with vertices as labelled, and suppose to the contrary that there is some graph G such that I (G) ∼ = T.
To begin, we determine the vertices of the two induced C 4 's of T. Immediately from Proposition 3.6, the vertices of X, A 1 , B 1 , and B 2 are as labelled in Figure 29.Using a second application of Proposition 3.6, A 2 differs from A 1 in exactly one position, different from B 2 .Without loss of generality, say, A 1 Then again by the proposition, Y = {y 1 , y 2 , y 3 , x 4 . . ., x k } as in Figure 29.
We now construct the vertices of D 1 through a series of claims: (i) x 3 is not in D 1 .
From Lemma 3.3, D 1 differs from X in one vertex, different from both A 1 and B 1 ; moreover, x 1 and x 2 are in D 1 .Notice that {x 4 , x 5 , . . ., x k } ⊆ D 1 : suppose to the contrary that, say, x 4 / ∈ D 1 , so that X (ii) y 1 , y 2 , and y 3 are not in D 1 .
As above, y 1 and y 2 are not in D 1 , as D 1 is an independent set containing x 1 and x 2 .Suppose to the contrary that X ∼ D 2 .Thus, we cannot build a set D 2 such that |D 2 ∩ Y | ≤ 2 as required.We so conclude that no such graph G exists, and that T is not an i-graph.
Like the seven non-i-graph realizable theta graphs of Theorem 5.3, T is a minimal obstruction to a graph being an i-graph; every induced subgraph of T is a theta graph.

Maximal Planar Graphs
We conclude this section with a result to demonstrate that certain planar graphs are igraphs and α-graphs.Our proof uses the following three known results.In our proof of the following theorem, we consider a graph G that is cubic, 3-connected, bipartite, and planar.We then examine its dual G, and how those specific properties of G translate to G.Then, we construct the complement of G, which we refer to as H.We claim that H is a seed graph of G; that is, I (H) contains an induced copy of G.
(b) is a theta graph, as illustrated by the seed graph G in Figure 2(a).The i-sets of G and their adjacencies are overlaid on H in Figure 2(b).Proposition 3.8 [2, 9] The house graph H is an i-graph.a b c d e (a) A graph G such that I (G) = H.{a, e} {a, d} {b, d} {b, e} {a, c} (b) The house graph H with i-sets of G.

Figure 2 :
Figure 2: The graph G for Proposition 3.8 with I (G) = H.

Figure 3 :
Figure 3: The complement and line graphs complement of the well-covered house graph H.

Theorem 7 . 2 [ 3 ,
Theorem 4.6] A cubic graph is 3-connected if and only if it is 3-edge connected.Theorem 7.3 [11] A connected planar graph G is bipartite if and only if its dual G is Eulerian.Theorem 7.4 [6, 10] A maximal planar graph G of order at least 3 has χ(G) = 3 if and only if G is Eulerian.

Theorem 7 . 5 Problem 6
Every cubic 3-connected bipartite planar graph is an i-graph and an α-graph.Proof.Let G be a cubic 3-connected bipartite planar graph and consider the dual G of G. Since G is bridgeless, G has no loops.Moreover, since G is 3-connected, Theorem 7.2 implies that no two edges separate G; hence, G has no multiple edges.Therefore, G is Problem 5 Determine the structure of i-graphs of various families of trees.For example, consider (a) caterpillars in which every vertex has degree 1 or 3, (b) spiders (K 1,r with each edge subdivided).Find more classes of i-graphs that are Hamiltonian, or Hamiltonian traceable.Problem 7 Suppose G 1 , G 2 , . . .are graphs such that I (G 1 ) ∼ = G 2 , I (G 2 ) ∼ = G 3 , I (G 3 ) ∼ = G 4 ,. . . .Under which conditions does there exist an integer k such that I (G k ) ∼ = G 1 ?As a special case of Problem 7, note that for any n ≥ 1, I (K n ) ∼ = K n , and that for k ≡ 2 (mod 3), I (C k ) ∼ = C k .Problem 8 Characterize the graphs G for which I (G) ∼ = G.