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Graphs and their associated inverse semigroups
Discrete Mathematics 340 (2017) 2408–2414
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Graphs and their associated inverse semigroups Tien Chih a , Demitri Plessas b, * a b
Montana State University-Billings, 1500 University Drive, Billings, MT 59101, USA Northeastern State University, 611 N. Grand Ave, Tahlequah, OK 74464, USA
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Article history: Received 6 June 2015 Received in revised form 4 April 2017 Accepted 18 May 2017
Keywords: Inverse semigroup Graph automorphism Graph symmetry Semigroup ideal Algebraic graph theory
a b s t r a c t Directed graphs have long been used to gain an understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Frucht’s Theorem. We investigate two inverse semigroups defined over undirected graphs constructed from the notions of subgraph and vertex set induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these inverse semigroups. We prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism allowing for an algebraic restatement of the Edge Reconstruction Conjecture. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Much of the theory linking semigroups to graphs have been in the guise of directed graphs [2,8,10,11,15,18]. However, undirected graphs have rich internal symmetries for which groups are too coarse an algebraic structure to distinguish. This has lead to the notions of distinguishing number [1] and fixing number [6]. Furthermore, local symmetry in the form of subgraph embeddings has been used(famously by Lovász to solve the edge reconstruction conjecture [7] for graphs with n ) n vertices and m edges where m ≥ 1/2 2 [14]. A possible algebraic structure to study local symmetry is an inverse semigroup. This leads us to investigate inverse semigroups on undirected graphs. In Section 2, we begin by defining inverse semigroups associated to undirected graphs to correspond to the ideas of subgraph symmetry and vertex set induced subgraph symmetry. These two inverse semigroups are linked to the edge reconstruction conjecture [7] and the vertex reconstruction conjecture [3]. These inverse semigroups are graph analogs of the inverse semigroup of sets [16] with a necessary restriction of partial monomorphism to partial isomorphism [5]. In Section 3, we characterize the idempotents and ideals for both inverse semigroups. In Section 4, we prove that the inverse semigroup corresponding to all subgraphs of a graph uniquely determines that graph up to isomorphism. Finally, in Section 5, we provide an algebraic restatement of the Edge Reconstruction Conjecture using inverse semigroups and consider the difficulties faced when attempting to form a similar restatement for the Vertex Reconstruction Conjecture. We will follow the notations of [4] for graph theory, and [9] for inverse semigroups. Specifically, given a graph G we denote the incidence function of G by ψG . We will only consider finite undirected graphs, but they are allowed to have multiple edges and loops. Given two graphs G and H, we define a graph isomorphism ϕ : G → H as a pair of bijections, ϕV : V (G) → V (H) and ϕE : E(G) → E(H) such that ψG (e) = uv if and only if ψH (ϕE (e)) = ϕV (u)ϕV (v ) for all vertices u and v of G. We allow ∅ to be considered a graph without vertices or edges and µ0 : ∅ → ∅ to be a valid graph isomorphism, where the bijections between the empty vertex sets and empty edge sets are empty maps.
*
Corresponding author. E-mail addresses: [email protected] (T. Chih), [email protected] (D. Plessas).
http://dx.doi.org/10.1016/j.disc.2017.05.008 0012-365X/© 2017 Elsevier B.V. All rights reserved.
T. Chih, D. Plessas / Discrete Mathematics 340 (2017) 2408–2414
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Fig. 1. A graph and two subgraph isomorphisms for G = ({v1 , v2 }, {e1 , e2 }).
2. Inverse semigroups constructed from graph symmetry We will start with the most general inverse semigroup associated to all subgraphs of a graph G. This is an inverse semigroup defined on the set of all partial isomorphisms of the subgraphs of G, including the empty graph, where the operation is a modified composition. Example 2.1. For an example of this operation, consider the graph G in Fig. 1 consisting of vertex set {v1 , v2 } with multiple edges e1 and e2 both incident to v1 and v2 . Define subgraph isomorphism ϕ : ({v1 , v2 }, {e1 }) → ({v1 , v2 }, {e1 }) by setting both ϕV and ϕE to be identity functions. Define subgraph isomorphism γ : ({v1 , v2 }, {e2 }) → ({v1 , v2 }, {e1 }) by setting γV to be the identity function and setting γE (e2 ) = e1 . Notice that the intersection of image of ϕE and the domain of γE is empty. Hence, γ ϕ is the identity subgraph isomorphism from ({v1 , v2 }, ∅) to itself. However, as the intersection of the image of γE and the domain of ϕE is {e1 }, ϕγ is the subgraph isomorphism from ({v1 , v2 }, {e2 }) to ({v1 , v2 }, {e1 }) where ϕγV is the identity function and ϕγE (e2 ) = e1 . Definition 2.2. Let G be a graph. We define the full inverse semigroup of G, denoted Fisg(G), to be the collection of all graph isomorphisms ϕ : H → J where H and J are subgraphs of G. We define the operation to be composition of isomorphisms. We now define the operation, denoted by multiplication, for this semigroup. For γ , ϕ ∈ Fisg(G) we define γ ϕ : ϕ −1 (Dom (γ )) → γ (Im(ϕ )) to be γ ◦ ϕ|ϕ −1 (Dom(γ )) , where ϕ|ϕ −1 (Dom(γ )) : ϕ −1 (Dom(γ )) → Dom(γ ) is the restriction graph isomorphism with vertex map ϕV |ϕ −1 (Dom(γV )) : V (ϕ −1 (Dom(γ ))) → V (Dom(γ )) and edge map ϕE |ϕ −1 (Dom(γE )) : E(ϕ −1 (Dom (γ ))) → E(Dom(γ )). In the case that the image of ϕ and the domain of γ are disjoint, γ ϕ will be µ0 , the identity isomorphism of the empty set graph. It is easy to see that γ ϕ is an isomorphism of subgraphs, and the operation is well defined. The operation of Fisg(G) is associative, and for any subgraph isomorphism ϕ , ϕϕ −1 ϕ = ϕ . Hence, Fisg(G) forms an inverse semigroup under the operation defined above. As we will see in Section 5, Fisg(G) is directly connected to the edge reconstruction conjecture. We obtain an analogous connection for the vertex reconstruction conjecture if we instead consider an inverse semigroup associated to subgraphs of G induced by vertex sets. Definition 2.3. Let G be a graph. We define the induced inverse semigroup of G, denoted Iisg(G), to be the collection of all graph isomorphisms ϕ : H → J where H and J are vertex subset induced subgraphs of G. We define the operation to be composition of isomorphisms. We then define the semigroup operation in the same way as for Fisg(G). Note that the intersection of two vertex induced subgraphs is a vertex induced subgraph when the empty vertex set and edge set graph is considered as the induced subgraph on ∅. Hence, Iisg(G) is also an inverse semigroup under the operation inherited from Fisg(G). 3. Idempotent and ideal structure In this section, we concern ourselves with the idempotents and ideals of our inverse semigroups. We categorize the idempotents and ideals of our inverse semigroups. Lemma 3.1. If G is a graph, then e ∈ Fisg(G) is idempotent if and only if there exists a subgraph H of G with e = idH , the identity automorphism. Proof. (⇐) For every subgraph H, idH is idempotent. (⇒) Let e ∈ Fisg(G) be an idempotent. As ee = e, Dom(e) = Im(e) and e is an automorphism of a subgraph H = Dom(e). As Aut(H) is a group, and the only idempotent of a group is the identity, e = idH . □ For any graph G, Iisg(G) is a subsemigroup of Fisg(G). Thus, the idempotents are the identities of vertex induced subgraphs of G. We then note that the semilattice of idempotents of Fisg(G) is isomorphic to the semilattice of subgraphs of G, and similarly the semilattice of idempotents of Iisg(G) is isomorphic to the semilattice of induced subgraphs of G (see Figs. 2 and 3). We now discuss ideals of a semigroup. Given a semigroup S and I ⊆ S, we say I is an ideal of S if IS , SI ⊆ I. This is of particular importance since ideals of a semigroup induce an equivalence relation which leads to the construction of a quotient
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Fig. 2. The idempotent lattice of Fisg(K2 ) when taken as identities of the given subgraphs.
Fig. 3. The idempotent lattice of Iisg(K2 ) when taken as identities of the given subgraphs.
semigroup via the Rees Factor Theorem [9,17]. Since each ideal I of S gives rise to a factor semigroup, and consequently a kernel of a semigroup homomorphism, one naturally wishes to classify all such ideals for the inverse semigroups of graphs and consider their structure. Recall that given a semigroup S, any union or intersection of ideals of S is also an ideal of S. Also, given a semigroup S and a ∈ S, we denote ⟨a⟩ to be the intersection of all ideals containing a. We call ⟨a⟩ the ideal generated by a. Equivalently, ⟨a⟩ = SaS [9]. Since µ0 , the empty map, is in Fisg(G), given any a ∈ Fisg(G), µ0 a = aµ0 = µ0 ∈ ⟨a⟩. Thus, the empty map is always an element of an ideal generated by any element of Fisg(G). We then classify the elements of ⟨a⟩ for a ∈ Fisg(G). Lemma 3.2. If G is a graph and a ∈ Fisg(G) where a : H → K , then {ϕ : L → M | L, M are isomorphic to a subgraph of H } is an ideal. Proof. Let α ∈ Fisg(G), and define Ia = {ϕ : L → M | L, M are isomorphic to a subgraph of H }. We consider α a. If Im(a) ∩ Dom(α ) = ∅ then α a is the empty map and is in Ia since the empty graph is a subgraph of H. Otherwise α a = α|Im(a) . Notice that Im(a) ∩ Dom(α ) is a subgraph of Im(a). Since Im(a) is isomorphic to H, Im(a) ∩ Dom(α ) is isomorphic to a subgraph of H and α a is an isomorphism of subgraphs isomorphic to a subgraph of H. Similarly, consider aα . If Im(α ) ∩ Dom(a) = ∅ then aα is the empty map and is in Ia . Otherwise aα = a|Im(α ) . Notice that Im(α ) ∩ Dom(a) is a subgraph of H. Thus, the domain α a is a subgraph of H and α a is an isomorphism of subgraphs isomorphic to a subgraph of H. □ Lemma 3.3. Let G be a graph and a ∈ Fisg(G) where a : H⋂→ K . If {ϕ : L → M | L, M are isomorphic to a subgraph of H }, then {ϕ : L → M | L, M are isomorphic to a subgraph of H } = Ii where Ii is an ideal of Fisg(G) containing a. ⋂ Proof. Let α ∈ Fisg(G), and define Ia as in Lemma 3.2. Since Ia is an ideal containing a, Ia ⊇ Ii . To show the other containment, we let I be an ideal containing a. Let L′ be a subgraph of H. Thus, idL′ ∈ Fisg(G), and aidL′ ∈ I. Since Im(idL′ ) is L′ , aidL′ = a|L′ and aidL′ is an isomorphism from L′ to an isomorphic subgraph of K , let us call this subgraph L′′ . Given any subgraph of G M isomorphic to L′ , there is an isomorphism ϕ : L′ → M. But since aidL′ is an isomorphism, it is invertible and moreover aidL′ ∈ I. Thus, ϕ (aidL′ )−1 (aidL′ ) ∈ I and ϕ ∈ I. Then, consider an induced subgraph of G L isomorphic to L′ . There is then an isomorphism δ : L → L′ . Since Im(δ ) = L′ = Dom(ϕ ), thus ϕδ is an isomorphism, where ϕδ : L → M. Since ϕ ∈ I , ϕδ ∈ I. Thus, given any L, M isomorphic to a subgraph of H, there is an isomorphism ϕδ ∈ I , ϕδ : L → M. So⋂given any isomorphism γ : L → M, γ (ϕδ )−1 (ϕδ ) ∈ I and γ ∈ I. Then, Ia ⊆ I. Since I was arbitrarily chosen, Ia ⊆ Ii and so ⋂ Ia = Ii . □
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Proposition 3.4. ⟨a⟩ = {ϕ ∈ S : Dom(ϕ ) ∼ = Dom(a)}. Proof. This is a consequence of Lemmas 3.2 and 3.3. □ We notice that an ideal generated by a single element of Fisg(G) is completely determined by the domain subgraph of that element. Since ideals are closed under unions and intersections, it is easy to see that any ideal is the union of all the principle ideals of its elements. One can see that an ideal is best understood by the subgraphs which constitute the domains of these functions. In order to deal with these subgraphs in a clear way, we introduce the notion of a basis. Definition 3.5. Let G be a graph and let I be an ideal of Fisg(G). Let B be a family of subgraphs of G, such that given any a : H → K in I, H is isomorphic to a subgraph of B for some B ∈ B. Moreover, given any B ∈ B, there is an α ∈ I where B is the domain of α . If such a family B exists, we call B a generating set of subgraphs of I. If B is minimal, we say B is a basis of I. We can think of a basis of an ideal I either as a minimal collection of subgraphs whose subgraphs are domains of elements if I, or we can think of the elements of B as the isomorphism classes of maximal subgraphs that are domains of elements of I. The term basis naturally invokes notions of minimality as well as uniqueness. We prove these essential properties of the basis. Lemma 3.6. Let G be a graph and let I be an ideal for Fisg(G). If C is a generating set of subgraphs for I, then there is a B ⊆ C such that B is a basis for I. Proof. If |C | = 1, then C = {C1 } is a singleton and is clearly minimal. So we consider the case where |C | > 1. If C is minimal with respect to being a generating set of graphs for I, then we let B = C and we are done. Otherwise there is a Cj , without loss of generality Cn where C \ {Cn } remains a generating set of graphs for I, and since |C | is finite, this process will terminate with a minimal generating subset. □ Corollary 3.7. Given a graph G and I an ideal of Fisg(G), I has a basis B. Proof. We notice that given any ideal I of Fisg(G), the set C = {Dom(a) : a ∈ I } is a generating set of subgraphs of I. Then, by Lemma 3.6, we can obtain a subset B of C that is a basis. □ Proposition 3.8. Let G be a graph and let I , J be ideals of Fisg(G) where I ⊆ J. If B is a generating set for J, then there is a basis for I, C where for each C ∈ C there is a B ∈ B where C isomorphic to a subgraph of some B. Proof. Let C be a basis for I. For each C C , there is an a ∈ I , a : C → K . Thus, C is the induced subgraph of H ′ where H ′ is a subgraph of G and there is a b ∈ J where b : H ′ → K ′ . Since B is a generating set for J, there is a B ∈ Bs where H ′ is isomorphic to a subgraph of B. Thus, C is isomorphic to a subgraph of B. □ Proposition 3.9. If G is a graph, and I , J are ideals of Fisg(G) with bases B, C , respectively, then B ∪ C is a generating set of graphs for I ∪ J. Proof. Let a ∈ I ∪ J. Either a ∈ I or a ∈ J. If a ∈ I then a : H → K where H is isomorphic to a subgraph of Bi ∈ B. Otherwise, if a ∈ J then a : M → L where M is isomorphic to a subgraph of Cj ∈ C . Either way, the domain of a is isomorphic to a subgraph of an element of B ∪ C . □ Theorem 3.10. If G is a graph, and I is an ideal of Fisg(G) with bases B and C , then there is a bijection Ψ : B → C such that for each B ∈ B it follows that B ∼ = Ψ (B). Proof. Let B = {B1 , . . . , Bm } and C = {C1 , . . . , Cn } be bases of I. Suppose that Bi is not isomorphic to any Cj ∈ C . Then, there must be a Ck where Bi is isomorphic to a subgraph of Ck . Since Ck ∈ C , Ck is isomorphic to the domain of some a ∈ I. Thus, there is a Bℓ where Ck is isomorphic to a subgraph of Bℓ . But, then Bi is isomorphic to a subgraph of Bℓ , contradicting the minimality of B. Thus, given each Bi ∈ B, there is a C ∈ C such that Bi ∼ = C , and we may define Ψ : B → C . □ Proposition 3.11. If H1 , . . . , Hn is a collection of subgraphs of G where no Hi is a subgraph of Hj when i ̸ = j, then has basis B = {H1 , . . . , Hn }.
⋃n
i=1
⟨idHi ⟩
Proof. Note that ⟨idHi ⟩ contains a subgraph isomorphism whose domain is H ⋃i .nThus, ⟨idHi ⟩ is the collection of all subgraph isomorphisms between any graphs isomorphic to a subgraph of Hi . Let I := i=1 ⟨idHi ⟩. Notice that if given any a ∈ I, then a ∈ ⟨idHi ⟩ for some i. Thus, the domain of a is isomorphic to a subgraph of Hi . Thus, B is a generating set for I. But since no Hi is a subgraph of any Hj , i ̸ = j and each idHi ∈ I, B is minimal since removing any Hi would contradict idHi ∈ I. □ This collection of results show that each ideal is determined exactly by a basis. Moreover, any collection of non-isomorphic subgraphs of G which pairwise non-monomorphic correspond to a basis of some I an ideal of Fisg(G).
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Fig. 4. A graph G and a basis of an ideal.
Example 3.12. Consider the graph ⎧ G in Fig. 4: Let us suppose we let B =
⎪ ⎨
b
, a
c
⎪ ⎩
⎫ ⎪ ⎬
.
⎪ ⎭
Since these subgraphs are pairwise non-monomorphic, B is the basis of an ideal I, which consists of the identity isomorphism on the subgraph: b the isomorphisms between c , a
a
c
b, b
,
including automorphisms; the isomorphisms between c , a
a
c
b, b
,
including automorphisms; and finally the isomorphisms and automorphisms between a, b, c . Also, notice that
B2 =
⎧ ⎪ ⎨ ⎪ ⎩
b
, a
b
⎫ ⎪ ⎬ ⎪ ⎭
, B3 =
⎧ ⎪ ⎨
b
, b
⎪ ⎩
c
⎫ ⎪ ⎬ ⎪ ⎭
are likewise basis of the same ideal, and that the elements of B2 , B3 are isomorphic to the elements of B. As we can see, it is much simpler to describe an ideal in terms of a basis than its individual elements. By Corollary 3.7, we know each ideal admits a basis and so finding the basis is just a matter of identifying the maximal domains of your ideal elements. Also, these results show that identifying ideals is as simple as identifying a set of pairwise non-monomorphic subgraphs as we did in Example 3.12, and that finding distinct ideals is just a matter of identifying such sets whose elements are non-isomorphic to each other. 4. Graph characterization by inverse semigroups We now consider the question of characterization. We show that Fisg(G) characterizes G. We then show this characterization will not hold in Iisg(G) and give an infinite class of counterexamples. Theorem 4.1. For G and H graphs, Fisg(G) ∼ = Fisg(H) if and only if G ∼ = H. Proof. Let Φ : Fisg(G) → Fisg(H) be a semigroup isomorphism. Let v ∈ V (G), and idv : {v} → {v} be the identity subgraph isomorphism. As idv is an idempotent, Φ (idv ) is an idempotent. By Lemma 3.1, Φ (idv ) is a subgraph automorphism of H. Furthermore, as the lattice structure of idempotents is preserved by the semigroup isomorphism, Φ (idv ) corresponds to a single vertex subgraph of H as Φ (idv ) covers the empty map in the idempotent lattice. Thus, we define ϕV : V (G) → V (H) by ϕV (v ) = Φ (idv )(v ). The inverse semigroup isomorphism, Φ −1 , allows us to similarly define ϕV−1 . Hence, ϕV is a bijection. To construct an edge mapping, we must consider loops and edges separately. Let e ∈ E(G) be a loop, and E be the subgraph of e and its incident vertex, v . The idempotent Φ (idE ) only covers Φ (idv ). Thus, Φ (idE ) is the identity of a subgraph of H consisting of an edge and its incident vertex Φ (idv )(v ). Thus, Φ (idE )(e) is a loop in E(H). Now, let e ∈ E(G) be a non-loop edge and E be the subgraph of e and its two incident vertices u and v . The idempotent Φ (idE ) covers the join of Φ (idu ) and Φ (idv ), and it does not cover any other single vertex identities. Hence, Φ (idE ) is a subgraph consisting of an edge and two incident vertices, namely Φ (idu )(u) and Φ (idv )(v ). Then, Φ (idE )(e) is an edge of H.
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Thus, for e ∈ E(G), we define ϕE : E(G) → E(H) by ϕE (e) = Φ (idE )(e) for E the subgraph consisting of e and its incident vertices (or vertex if it is a loop). We can use a similar argument to define ϕE−1 using Φ −1 . Hence, ϕE is a bijection. Let e ∈ E(G) incident to u, v ∈ V (G). Then, as Φ (idE )(E) is a subgraph of H with vertices Φ (idu )(u) and Φ (idv )(v ), ϕE (e) = Φ (idE )(e) is incident to ϕV (u) = Φ (idu )(u) and ϕV (v ) = Φ (idv )(v ). Thus, ϕE preserves incidence. Hence, ϕ = (ϕV , ϕE ) is a graph isomorphism and G ∼ = H. The converse is a direct consequence of the graph isomorphism. □ We obtain a similar but weaker result for Iisg(G). Proposition 4.2. If G is a simple graph and G is its complement graph, then Iisg(G) ∼ = Iisg(G). Proof. As isomorphisms preserve both adjacency and non-adjacency in simple graphs, an isomorphism of vertex induced subgraphs of a simple graph induces an isomorphism of the subgraphs of its complement on the same vertex sets. Thus, they will have isomorphic induced subgraph inverse semigroups. □ Example 4.3. For Kn and Kn the empty edge graph on n vertices Iisg(Kn ) ∼ = Iisg(Kn ). 5. Connections to the reconstruction conjectures In this section, we exhibit applications of our inverse semigroups to the Vertex Reconstruction Conjecture [12] and the Edge Reconstruction Conjecture [19]. Given a graph G with n vertices and allow {G1 , G2 , . . . , Gn } to denote the collection of vertex-deleted subgraphs of G. Suppose H is a graph with n vertices where the vertex-deleted subgraphs of H may be denoted {H1 , H2 , . . . , Hn }, such that Gi ∼ = Hi for all 1 ≤ i ≤ n. If for each such H it follows that G ∼ = H, then we say G is vertex-reconstructible. Similarly, let G be a graph withm edges, and Gi denote edge-deletions of G, respectively, for 1 ≤ i ≤ m. Let H be a graph with m edges where the edge-deletions may be denoted {H1 , H2 , . . . , Hm }, such that Gi ∼ = Hi for all 1 ≤ i ≤ m. If for each such H it follows that G ∼ = H, we say that G is edge-reconstructible. The Reconstruction Conjectures state that all finite graphs with at least 3 vertices or graphs with at least 4 edges are vertex and edge reconstructible, respectively. In essence, these conjectures posit that with finitely many exceptions, all finite graphs are determined by their subgraphs. By Lemma 3.1, we see that the idempotents of our inverse semigroups are exactly identities of subgraphs. Thus, it is natural to explore the connection between our inverse semigroups and these conjectures. Given a semigroup S, and an idempotent e, we say eSe is a local submonoid [13]. In particular, given a graph G, every ˆ is a subgraph of G. Since every element of id ˆ Fisg(G)id ˆ is a subgraph idempotent of Fisg(G) is of the form idGˆ where G G G ˆ and all such subgraph isomorphisms would be contained in isomorphism whose domain and co-domain are subgraphs of G, ˆ This leads to the following definition and theorem. idGˆ Fisg(G)idGˆ , it is easy to see that idGˆ Fisg(G)idGˆ = Fisg(G). Definition 5.1. Given a graph G, let {e1 , e2 , . . . , ek } denote the collection of idempotents of Fisg(G) covered by idG . Let H be a graph where the idempotents of H covered by idH may be denoted {ˆe1 , eˆ 2 , . . . , eˆ j }, such that j = k and ei Fisg(G)ei ∼ = eˆ i Fisg(H)eˆ i for all 1 ≤ i ≤ k. If for each such H it follows that Fisg(G) ∼ = Fisg(H), we say that Fisg(G) is full lattice determined. Similarly, we define induced lattice determined for Iisg(G). Theorem 5.2. A graph G with no isolated vertices is edge-reconstructible if and only if Fisg(G) is full lattice determined. Proof. Let {e1 , e2 , . . . , ek } denote the collection of idempotents of Fisg(G) covered by idG . By Lemma 3.1, these idempotents covered by idG are identity automorphisms of subgraphs of G. In particular, since they are covered by idG , and G has no isolated vertices, we observe that ei = idGi , where Gi is an edge-deleted subgraph of G for all 1 ≤ i ≤ k. Suppose G is edge-reconstructible. Given any graph H where the idempotents of H covered by idH may be denoted {ˆe1 , eˆ 2 , . . . , eˆ j }, such that j = k and ei Fisg(G)ei ∼ = eˆ i Fisg(H)eˆ i for all 1 ≤ i ≤ k, we see that for any idempotent ei , ei Fisg(G)ei = Fisg(Gi ) ∼ = Fisg(Hi ) = eˆ i Fisg(H)eˆ i , where Gi , Hi are edge deleted subgraphs of G and H, respectively. By Theorem 4.1, Gi ∼ = Hi for all 1 ≤ i ≤ k. This holds for each idempotent ei , and each such idempotent corresponds to an edge-deleted subgraph of G. Since G is edge-reconstructible, G ∼ = H. By Theorem 4.1, Fisg(G) ∼ = Fisg(H). Therefore, Fisg(G) is full lattice determined. Similarly, suppose Fisg(G) is full lattice determined. Given any graph H where the edge-deleted subgraphs of H may be denoted {H1 , H2 , . . . , Hm }, such that m = k and Gi ∼ = Hi for all 1 ≤ i ≤ k, we see that by Theorem 4.1, Fisg(Gi ) = Fisg(Hi ) for all 1 ≤ i ≤ k. Since this is true for each edge-deleted subgraph, and Fisg(G) is full lattice determined, Fisg(G) ∼ = Fisg(H) Thus, by Theorem 4.1, G ∼ = H, and G is edge-reconstructible. □ Notice that the idempotents covered by idG in Iisg(G) are vertex-deleted subgraphs of G. It is also easy to see that vertex deleted subgraphs who are isomorphic will have isomorphic induced inverse semigroups. However, Proposition 4.2 does not provide the converse. Thus, we do not have a result as strong as Theorem 5.2. Hence, we propose the following question: Problem 1. Find necessary and sufficient properties for Iisg(G) so that G is vertex-reconstructible.
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Acknowledgments The authors are grateful to the referees whose helpful suggestions greatly improved this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
M.O. Albertson, K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1) (1996) R18. C.J. Ash, T.E. Hall, Inverse semigroups on graphs, Semigroup Forum 11 (1) (1975) 140–145. J.A. Bondy, R.L. Hemminger, Graph reconstruction –a survey, J. Graph Theory 1 (3) (1977) 227–268. J.A. Bondy, U.S.R. Murty, Graph theory. 2008, Grad. Texts in Math. (2008). T. Chih, The groupoid of induced subgraphs, 2016, under review. C.R. Gibbons, J.D. Laison, Fixing numbers of graphs and groups, Electron. J. Combin. 16 (1) (2009) R39. F. Harary, On the reconstruction of a graph from a collection of subgraphs, in: Theory of Graphs and Its Applications (Proc. Sympos. Smolenice, 1963), 1964, pp. 47–52. D.G. Jones, M.V. Lawson, Graph inverse semigroups: their characterization and completion, J. Algebra 409 (2014) 444–473. A.V. Kelarev, Graph Algebras and Automata, Marcel Dekker, 2003. A.V. Kelarev, On Cayley graphs of inverse semigroups, Semigroup Forum 72 (3) (2006) 411–418. A.V. Kelarev, J. Ryan, J. Yearwood, Cayley graphs as classifiers for data mining: the influence of asymmetries, Discrete Math. 309 (17) (2009) 5360–5369. P. Kelly, A congurence theorem for trees, Pacific J. Math. 7 (1957) 961–968. M.V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific Publishing Company, 1999. L. Lovász, A note on the line reconstruction problem, in: Classic Papers in Combinatorics, Springer, 1987, pp. 451–452. S.W. Margolis, J.C. Meakin, E-unitary inverse monoids and the Cayley graph of a group presentation, J. Pure Appl. Algebra 58 (1) (1989) 45–76. G.B. Preston, Representations of inverse Semigroups, J. Lond. Math. Soc. 29 (1954). D. Rees, On semigroups, Proc. Cambridge Philos. Soc. 36 (1940). N. Sieben, Cayley color graphs of inverse semigroups and groupoids, Czechoslovak Math. J. 58 (3) (2008) 683–692. S. Ulam, A Collection of Mathematical Problems, Wiley, 1960.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Graphs and their associated inverse semigroups Tien Chih a , Demitri Plessas b, * a b
Montana State University-Billings, 1500 University Drive, Billings, MT 59101, USA Northeastern State University, 611 N. Grand Ave, Tahlequah, OK 74464, USA
article
info
Article history: Received 6 June 2015 Received in revised form 4 April 2017 Accepted 18 May 2017
Keywords: Inverse semigroup Graph automorphism Graph symmetry Semigroup ideal Algebraic graph theory
a b s t r a c t Directed graphs have long been used to gain an understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Frucht’s Theorem. We investigate two inverse semigroups defined over undirected graphs constructed from the notions of subgraph and vertex set induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these inverse semigroups. We prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism allowing for an algebraic restatement of the Edge Reconstruction Conjecture. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Much of the theory linking semigroups to graphs have been in the guise of directed graphs [2,8,10,11,15,18]. However, undirected graphs have rich internal symmetries for which groups are too coarse an algebraic structure to distinguish. This has lead to the notions of distinguishing number [1] and fixing number [6]. Furthermore, local symmetry in the form of subgraph embeddings has been used(famously by Lovász to solve the edge reconstruction conjecture [7] for graphs with n ) n vertices and m edges where m ≥ 1/2 2 [14]. A possible algebraic structure to study local symmetry is an inverse semigroup. This leads us to investigate inverse semigroups on undirected graphs. In Section 2, we begin by defining inverse semigroups associated to undirected graphs to correspond to the ideas of subgraph symmetry and vertex set induced subgraph symmetry. These two inverse semigroups are linked to the edge reconstruction conjecture [7] and the vertex reconstruction conjecture [3]. These inverse semigroups are graph analogs of the inverse semigroup of sets [16] with a necessary restriction of partial monomorphism to partial isomorphism [5]. In Section 3, we characterize the idempotents and ideals for both inverse semigroups. In Section 4, we prove that the inverse semigroup corresponding to all subgraphs of a graph uniquely determines that graph up to isomorphism. Finally, in Section 5, we provide an algebraic restatement of the Edge Reconstruction Conjecture using inverse semigroups and consider the difficulties faced when attempting to form a similar restatement for the Vertex Reconstruction Conjecture. We will follow the notations of [4] for graph theory, and [9] for inverse semigroups. Specifically, given a graph G we denote the incidence function of G by ψG . We will only consider finite undirected graphs, but they are allowed to have multiple edges and loops. Given two graphs G and H, we define a graph isomorphism ϕ : G → H as a pair of bijections, ϕV : V (G) → V (H) and ϕE : E(G) → E(H) such that ψG (e) = uv if and only if ψH (ϕE (e)) = ϕV (u)ϕV (v ) for all vertices u and v of G. We allow ∅ to be considered a graph without vertices or edges and µ0 : ∅ → ∅ to be a valid graph isomorphism, where the bijections between the empty vertex sets and empty edge sets are empty maps.
*
Corresponding author. E-mail addresses: [email protected] (T. Chih), [email protected] (D. Plessas).
http://dx.doi.org/10.1016/j.disc.2017.05.008 0012-365X/© 2017 Elsevier B.V. All rights reserved.
T. Chih, D. Plessas / Discrete Mathematics 340 (2017) 2408–2414
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Fig. 1. A graph and two subgraph isomorphisms for G = ({v1 , v2 }, {e1 , e2 }).
2. Inverse semigroups constructed from graph symmetry We will start with the most general inverse semigroup associated to all subgraphs of a graph G. This is an inverse semigroup defined on the set of all partial isomorphisms of the subgraphs of G, including the empty graph, where the operation is a modified composition. Example 2.1. For an example of this operation, consider the graph G in Fig. 1 consisting of vertex set {v1 , v2 } with multiple edges e1 and e2 both incident to v1 and v2 . Define subgraph isomorphism ϕ : ({v1 , v2 }, {e1 }) → ({v1 , v2 }, {e1 }) by setting both ϕV and ϕE to be identity functions. Define subgraph isomorphism γ : ({v1 , v2 }, {e2 }) → ({v1 , v2 }, {e1 }) by setting γV to be the identity function and setting γE (e2 ) = e1 . Notice that the intersection of image of ϕE and the domain of γE is empty. Hence, γ ϕ is the identity subgraph isomorphism from ({v1 , v2 }, ∅) to itself. However, as the intersection of the image of γE and the domain of ϕE is {e1 }, ϕγ is the subgraph isomorphism from ({v1 , v2 }, {e2 }) to ({v1 , v2 }, {e1 }) where ϕγV is the identity function and ϕγE (e2 ) = e1 . Definition 2.2. Let G be a graph. We define the full inverse semigroup of G, denoted Fisg(G), to be the collection of all graph isomorphisms ϕ : H → J where H and J are subgraphs of G. We define the operation to be composition of isomorphisms. We now define the operation, denoted by multiplication, for this semigroup. For γ , ϕ ∈ Fisg(G) we define γ ϕ : ϕ −1 (Dom (γ )) → γ (Im(ϕ )) to be γ ◦ ϕ|ϕ −1 (Dom(γ )) , where ϕ|ϕ −1 (Dom(γ )) : ϕ −1 (Dom(γ )) → Dom(γ ) is the restriction graph isomorphism with vertex map ϕV |ϕ −1 (Dom(γV )) : V (ϕ −1 (Dom(γ ))) → V (Dom(γ )) and edge map ϕE |ϕ −1 (Dom(γE )) : E(ϕ −1 (Dom (γ ))) → E(Dom(γ )). In the case that the image of ϕ and the domain of γ are disjoint, γ ϕ will be µ0 , the identity isomorphism of the empty set graph. It is easy to see that γ ϕ is an isomorphism of subgraphs, and the operation is well defined. The operation of Fisg(G) is associative, and for any subgraph isomorphism ϕ , ϕϕ −1 ϕ = ϕ . Hence, Fisg(G) forms an inverse semigroup under the operation defined above. As we will see in Section 5, Fisg(G) is directly connected to the edge reconstruction conjecture. We obtain an analogous connection for the vertex reconstruction conjecture if we instead consider an inverse semigroup associated to subgraphs of G induced by vertex sets. Definition 2.3. Let G be a graph. We define the induced inverse semigroup of G, denoted Iisg(G), to be the collection of all graph isomorphisms ϕ : H → J where H and J are vertex subset induced subgraphs of G. We define the operation to be composition of isomorphisms. We then define the semigroup operation in the same way as for Fisg(G). Note that the intersection of two vertex induced subgraphs is a vertex induced subgraph when the empty vertex set and edge set graph is considered as the induced subgraph on ∅. Hence, Iisg(G) is also an inverse semigroup under the operation inherited from Fisg(G). 3. Idempotent and ideal structure In this section, we concern ourselves with the idempotents and ideals of our inverse semigroups. We categorize the idempotents and ideals of our inverse semigroups. Lemma 3.1. If G is a graph, then e ∈ Fisg(G) is idempotent if and only if there exists a subgraph H of G with e = idH , the identity automorphism. Proof. (⇐) For every subgraph H, idH is idempotent. (⇒) Let e ∈ Fisg(G) be an idempotent. As ee = e, Dom(e) = Im(e) and e is an automorphism of a subgraph H = Dom(e). As Aut(H) is a group, and the only idempotent of a group is the identity, e = idH . □ For any graph G, Iisg(G) is a subsemigroup of Fisg(G). Thus, the idempotents are the identities of vertex induced subgraphs of G. We then note that the semilattice of idempotents of Fisg(G) is isomorphic to the semilattice of subgraphs of G, and similarly the semilattice of idempotents of Iisg(G) is isomorphic to the semilattice of induced subgraphs of G (see Figs. 2 and 3). We now discuss ideals of a semigroup. Given a semigroup S and I ⊆ S, we say I is an ideal of S if IS , SI ⊆ I. This is of particular importance since ideals of a semigroup induce an equivalence relation which leads to the construction of a quotient
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Fig. 2. The idempotent lattice of Fisg(K2 ) when taken as identities of the given subgraphs.
Fig. 3. The idempotent lattice of Iisg(K2 ) when taken as identities of the given subgraphs.
semigroup via the Rees Factor Theorem [9,17]. Since each ideal I of S gives rise to a factor semigroup, and consequently a kernel of a semigroup homomorphism, one naturally wishes to classify all such ideals for the inverse semigroups of graphs and consider their structure. Recall that given a semigroup S, any union or intersection of ideals of S is also an ideal of S. Also, given a semigroup S and a ∈ S, we denote ⟨a⟩ to be the intersection of all ideals containing a. We call ⟨a⟩ the ideal generated by a. Equivalently, ⟨a⟩ = SaS [9]. Since µ0 , the empty map, is in Fisg(G), given any a ∈ Fisg(G), µ0 a = aµ0 = µ0 ∈ ⟨a⟩. Thus, the empty map is always an element of an ideal generated by any element of Fisg(G). We then classify the elements of ⟨a⟩ for a ∈ Fisg(G). Lemma 3.2. If G is a graph and a ∈ Fisg(G) where a : H → K , then {ϕ : L → M | L, M are isomorphic to a subgraph of H } is an ideal. Proof. Let α ∈ Fisg(G), and define Ia = {ϕ : L → M | L, M are isomorphic to a subgraph of H }. We consider α a. If Im(a) ∩ Dom(α ) = ∅ then α a is the empty map and is in Ia since the empty graph is a subgraph of H. Otherwise α a = α|Im(a) . Notice that Im(a) ∩ Dom(α ) is a subgraph of Im(a). Since Im(a) is isomorphic to H, Im(a) ∩ Dom(α ) is isomorphic to a subgraph of H and α a is an isomorphism of subgraphs isomorphic to a subgraph of H. Similarly, consider aα . If Im(α ) ∩ Dom(a) = ∅ then aα is the empty map and is in Ia . Otherwise aα = a|Im(α ) . Notice that Im(α ) ∩ Dom(a) is a subgraph of H. Thus, the domain α a is a subgraph of H and α a is an isomorphism of subgraphs isomorphic to a subgraph of H. □ Lemma 3.3. Let G be a graph and a ∈ Fisg(G) where a : H⋂→ K . If {ϕ : L → M | L, M are isomorphic to a subgraph of H }, then {ϕ : L → M | L, M are isomorphic to a subgraph of H } = Ii where Ii is an ideal of Fisg(G) containing a. ⋂ Proof. Let α ∈ Fisg(G), and define Ia as in Lemma 3.2. Since Ia is an ideal containing a, Ia ⊇ Ii . To show the other containment, we let I be an ideal containing a. Let L′ be a subgraph of H. Thus, idL′ ∈ Fisg(G), and aidL′ ∈ I. Since Im(idL′ ) is L′ , aidL′ = a|L′ and aidL′ is an isomorphism from L′ to an isomorphic subgraph of K , let us call this subgraph L′′ . Given any subgraph of G M isomorphic to L′ , there is an isomorphism ϕ : L′ → M. But since aidL′ is an isomorphism, it is invertible and moreover aidL′ ∈ I. Thus, ϕ (aidL′ )−1 (aidL′ ) ∈ I and ϕ ∈ I. Then, consider an induced subgraph of G L isomorphic to L′ . There is then an isomorphism δ : L → L′ . Since Im(δ ) = L′ = Dom(ϕ ), thus ϕδ is an isomorphism, where ϕδ : L → M. Since ϕ ∈ I , ϕδ ∈ I. Thus, given any L, M isomorphic to a subgraph of H, there is an isomorphism ϕδ ∈ I , ϕδ : L → M. So⋂given any isomorphism γ : L → M, γ (ϕδ )−1 (ϕδ ) ∈ I and γ ∈ I. Then, Ia ⊆ I. Since I was arbitrarily chosen, Ia ⊆ Ii and so ⋂ Ia = Ii . □
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Proposition 3.4. ⟨a⟩ = {ϕ ∈ S : Dom(ϕ ) ∼ = Dom(a)}. Proof. This is a consequence of Lemmas 3.2 and 3.3. □ We notice that an ideal generated by a single element of Fisg(G) is completely determined by the domain subgraph of that element. Since ideals are closed under unions and intersections, it is easy to see that any ideal is the union of all the principle ideals of its elements. One can see that an ideal is best understood by the subgraphs which constitute the domains of these functions. In order to deal with these subgraphs in a clear way, we introduce the notion of a basis. Definition 3.5. Let G be a graph and let I be an ideal of Fisg(G). Let B be a family of subgraphs of G, such that given any a : H → K in I, H is isomorphic to a subgraph of B for some B ∈ B. Moreover, given any B ∈ B, there is an α ∈ I where B is the domain of α . If such a family B exists, we call B a generating set of subgraphs of I. If B is minimal, we say B is a basis of I. We can think of a basis of an ideal I either as a minimal collection of subgraphs whose subgraphs are domains of elements if I, or we can think of the elements of B as the isomorphism classes of maximal subgraphs that are domains of elements of I. The term basis naturally invokes notions of minimality as well as uniqueness. We prove these essential properties of the basis. Lemma 3.6. Let G be a graph and let I be an ideal for Fisg(G). If C is a generating set of subgraphs for I, then there is a B ⊆ C such that B is a basis for I. Proof. If |C | = 1, then C = {C1 } is a singleton and is clearly minimal. So we consider the case where |C | > 1. If C is minimal with respect to being a generating set of graphs for I, then we let B = C and we are done. Otherwise there is a Cj , without loss of generality Cn where C \ {Cn } remains a generating set of graphs for I, and since |C | is finite, this process will terminate with a minimal generating subset. □ Corollary 3.7. Given a graph G and I an ideal of Fisg(G), I has a basis B. Proof. We notice that given any ideal I of Fisg(G), the set C = {Dom(a) : a ∈ I } is a generating set of subgraphs of I. Then, by Lemma 3.6, we can obtain a subset B of C that is a basis. □ Proposition 3.8. Let G be a graph and let I , J be ideals of Fisg(G) where I ⊆ J. If B is a generating set for J, then there is a basis for I, C where for each C ∈ C there is a B ∈ B where C isomorphic to a subgraph of some B. Proof. Let C be a basis for I. For each C C , there is an a ∈ I , a : C → K . Thus, C is the induced subgraph of H ′ where H ′ is a subgraph of G and there is a b ∈ J where b : H ′ → K ′ . Since B is a generating set for J, there is a B ∈ Bs where H ′ is isomorphic to a subgraph of B. Thus, C is isomorphic to a subgraph of B. □ Proposition 3.9. If G is a graph, and I , J are ideals of Fisg(G) with bases B, C , respectively, then B ∪ C is a generating set of graphs for I ∪ J. Proof. Let a ∈ I ∪ J. Either a ∈ I or a ∈ J. If a ∈ I then a : H → K where H is isomorphic to a subgraph of Bi ∈ B. Otherwise, if a ∈ J then a : M → L where M is isomorphic to a subgraph of Cj ∈ C . Either way, the domain of a is isomorphic to a subgraph of an element of B ∪ C . □ Theorem 3.10. If G is a graph, and I is an ideal of Fisg(G) with bases B and C , then there is a bijection Ψ : B → C such that for each B ∈ B it follows that B ∼ = Ψ (B). Proof. Let B = {B1 , . . . , Bm } and C = {C1 , . . . , Cn } be bases of I. Suppose that Bi is not isomorphic to any Cj ∈ C . Then, there must be a Ck where Bi is isomorphic to a subgraph of Ck . Since Ck ∈ C , Ck is isomorphic to the domain of some a ∈ I. Thus, there is a Bℓ where Ck is isomorphic to a subgraph of Bℓ . But, then Bi is isomorphic to a subgraph of Bℓ , contradicting the minimality of B. Thus, given each Bi ∈ B, there is a C ∈ C such that Bi ∼ = C , and we may define Ψ : B → C . □ Proposition 3.11. If H1 , . . . , Hn is a collection of subgraphs of G where no Hi is a subgraph of Hj when i ̸ = j, then has basis B = {H1 , . . . , Hn }.
⋃n
i=1
⟨idHi ⟩
Proof. Note that ⟨idHi ⟩ contains a subgraph isomorphism whose domain is H ⋃i .nThus, ⟨idHi ⟩ is the collection of all subgraph isomorphisms between any graphs isomorphic to a subgraph of Hi . Let I := i=1 ⟨idHi ⟩. Notice that if given any a ∈ I, then a ∈ ⟨idHi ⟩ for some i. Thus, the domain of a is isomorphic to a subgraph of Hi . Thus, B is a generating set for I. But since no Hi is a subgraph of any Hj , i ̸ = j and each idHi ∈ I, B is minimal since removing any Hi would contradict idHi ∈ I. □ This collection of results show that each ideal is determined exactly by a basis. Moreover, any collection of non-isomorphic subgraphs of G which pairwise non-monomorphic correspond to a basis of some I an ideal of Fisg(G).
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Fig. 4. A graph G and a basis of an ideal.
Example 3.12. Consider the graph ⎧ G in Fig. 4: Let us suppose we let B =
⎪ ⎨
b
, a
c
⎪ ⎩
⎫ ⎪ ⎬
.
⎪ ⎭
Since these subgraphs are pairwise non-monomorphic, B is the basis of an ideal I, which consists of the identity isomorphism on the subgraph: b the isomorphisms between c , a
a
c
b, b
,
including automorphisms; the isomorphisms between c , a
a
c
b, b
,
including automorphisms; and finally the isomorphisms and automorphisms between a, b, c . Also, notice that
B2 =
⎧ ⎪ ⎨ ⎪ ⎩
b
, a
b
⎫ ⎪ ⎬ ⎪ ⎭
, B3 =
⎧ ⎪ ⎨
b
, b
⎪ ⎩
c
⎫ ⎪ ⎬ ⎪ ⎭
are likewise basis of the same ideal, and that the elements of B2 , B3 are isomorphic to the elements of B. As we can see, it is much simpler to describe an ideal in terms of a basis than its individual elements. By Corollary 3.7, we know each ideal admits a basis and so finding the basis is just a matter of identifying the maximal domains of your ideal elements. Also, these results show that identifying ideals is as simple as identifying a set of pairwise non-monomorphic subgraphs as we did in Example 3.12, and that finding distinct ideals is just a matter of identifying such sets whose elements are non-isomorphic to each other. 4. Graph characterization by inverse semigroups We now consider the question of characterization. We show that Fisg(G) characterizes G. We then show this characterization will not hold in Iisg(G) and give an infinite class of counterexamples. Theorem 4.1. For G and H graphs, Fisg(G) ∼ = Fisg(H) if and only if G ∼ = H. Proof. Let Φ : Fisg(G) → Fisg(H) be a semigroup isomorphism. Let v ∈ V (G), and idv : {v} → {v} be the identity subgraph isomorphism. As idv is an idempotent, Φ (idv ) is an idempotent. By Lemma 3.1, Φ (idv ) is a subgraph automorphism of H. Furthermore, as the lattice structure of idempotents is preserved by the semigroup isomorphism, Φ (idv ) corresponds to a single vertex subgraph of H as Φ (idv ) covers the empty map in the idempotent lattice. Thus, we define ϕV : V (G) → V (H) by ϕV (v ) = Φ (idv )(v ). The inverse semigroup isomorphism, Φ −1 , allows us to similarly define ϕV−1 . Hence, ϕV is a bijection. To construct an edge mapping, we must consider loops and edges separately. Let e ∈ E(G) be a loop, and E be the subgraph of e and its incident vertex, v . The idempotent Φ (idE ) only covers Φ (idv ). Thus, Φ (idE ) is the identity of a subgraph of H consisting of an edge and its incident vertex Φ (idv )(v ). Thus, Φ (idE )(e) is a loop in E(H). Now, let e ∈ E(G) be a non-loop edge and E be the subgraph of e and its two incident vertices u and v . The idempotent Φ (idE ) covers the join of Φ (idu ) and Φ (idv ), and it does not cover any other single vertex identities. Hence, Φ (idE ) is a subgraph consisting of an edge and two incident vertices, namely Φ (idu )(u) and Φ (idv )(v ). Then, Φ (idE )(e) is an edge of H.
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Thus, for e ∈ E(G), we define ϕE : E(G) → E(H) by ϕE (e) = Φ (idE )(e) for E the subgraph consisting of e and its incident vertices (or vertex if it is a loop). We can use a similar argument to define ϕE−1 using Φ −1 . Hence, ϕE is a bijection. Let e ∈ E(G) incident to u, v ∈ V (G). Then, as Φ (idE )(E) is a subgraph of H with vertices Φ (idu )(u) and Φ (idv )(v ), ϕE (e) = Φ (idE )(e) is incident to ϕV (u) = Φ (idu )(u) and ϕV (v ) = Φ (idv )(v ). Thus, ϕE preserves incidence. Hence, ϕ = (ϕV , ϕE ) is a graph isomorphism and G ∼ = H. The converse is a direct consequence of the graph isomorphism. □ We obtain a similar but weaker result for Iisg(G). Proposition 4.2. If G is a simple graph and G is its complement graph, then Iisg(G) ∼ = Iisg(G). Proof. As isomorphisms preserve both adjacency and non-adjacency in simple graphs, an isomorphism of vertex induced subgraphs of a simple graph induces an isomorphism of the subgraphs of its complement on the same vertex sets. Thus, they will have isomorphic induced subgraph inverse semigroups. □ Example 4.3. For Kn and Kn the empty edge graph on n vertices Iisg(Kn ) ∼ = Iisg(Kn ). 5. Connections to the reconstruction conjectures In this section, we exhibit applications of our inverse semigroups to the Vertex Reconstruction Conjecture [12] and the Edge Reconstruction Conjecture [19]. Given a graph G with n vertices and allow {G1 , G2 , . . . , Gn } to denote the collection of vertex-deleted subgraphs of G. Suppose H is a graph with n vertices where the vertex-deleted subgraphs of H may be denoted {H1 , H2 , . . . , Hn }, such that Gi ∼ = Hi for all 1 ≤ i ≤ n. If for each such H it follows that G ∼ = H, then we say G is vertex-reconstructible. Similarly, let G be a graph withm edges, and Gi denote edge-deletions of G, respectively, for 1 ≤ i ≤ m. Let H be a graph with m edges where the edge-deletions may be denoted {H1 , H2 , . . . , Hm }, such that Gi ∼ = Hi for all 1 ≤ i ≤ m. If for each such H it follows that G ∼ = H, we say that G is edge-reconstructible. The Reconstruction Conjectures state that all finite graphs with at least 3 vertices or graphs with at least 4 edges are vertex and edge reconstructible, respectively. In essence, these conjectures posit that with finitely many exceptions, all finite graphs are determined by their subgraphs. By Lemma 3.1, we see that the idempotents of our inverse semigroups are exactly identities of subgraphs. Thus, it is natural to explore the connection between our inverse semigroups and these conjectures. Given a semigroup S, and an idempotent e, we say eSe is a local submonoid [13]. In particular, given a graph G, every ˆ is a subgraph of G. Since every element of id ˆ Fisg(G)id ˆ is a subgraph idempotent of Fisg(G) is of the form idGˆ where G G G ˆ and all such subgraph isomorphisms would be contained in isomorphism whose domain and co-domain are subgraphs of G, ˆ This leads to the following definition and theorem. idGˆ Fisg(G)idGˆ , it is easy to see that idGˆ Fisg(G)idGˆ = Fisg(G). Definition 5.1. Given a graph G, let {e1 , e2 , . . . , ek } denote the collection of idempotents of Fisg(G) covered by idG . Let H be a graph where the idempotents of H covered by idH may be denoted {ˆe1 , eˆ 2 , . . . , eˆ j }, such that j = k and ei Fisg(G)ei ∼ = eˆ i Fisg(H)eˆ i for all 1 ≤ i ≤ k. If for each such H it follows that Fisg(G) ∼ = Fisg(H), we say that Fisg(G) is full lattice determined. Similarly, we define induced lattice determined for Iisg(G). Theorem 5.2. A graph G with no isolated vertices is edge-reconstructible if and only if Fisg(G) is full lattice determined. Proof. Let {e1 , e2 , . . . , ek } denote the collection of idempotents of Fisg(G) covered by idG . By Lemma 3.1, these idempotents covered by idG are identity automorphisms of subgraphs of G. In particular, since they are covered by idG , and G has no isolated vertices, we observe that ei = idGi , where Gi is an edge-deleted subgraph of G for all 1 ≤ i ≤ k. Suppose G is edge-reconstructible. Given any graph H where the idempotents of H covered by idH may be denoted {ˆe1 , eˆ 2 , . . . , eˆ j }, such that j = k and ei Fisg(G)ei ∼ = eˆ i Fisg(H)eˆ i for all 1 ≤ i ≤ k, we see that for any idempotent ei , ei Fisg(G)ei = Fisg(Gi ) ∼ = Fisg(Hi ) = eˆ i Fisg(H)eˆ i , where Gi , Hi are edge deleted subgraphs of G and H, respectively. By Theorem 4.1, Gi ∼ = Hi for all 1 ≤ i ≤ k. This holds for each idempotent ei , and each such idempotent corresponds to an edge-deleted subgraph of G. Since G is edge-reconstructible, G ∼ = H. By Theorem 4.1, Fisg(G) ∼ = Fisg(H). Therefore, Fisg(G) is full lattice determined. Similarly, suppose Fisg(G) is full lattice determined. Given any graph H where the edge-deleted subgraphs of H may be denoted {H1 , H2 , . . . , Hm }, such that m = k and Gi ∼ = Hi for all 1 ≤ i ≤ k, we see that by Theorem 4.1, Fisg(Gi ) = Fisg(Hi ) for all 1 ≤ i ≤ k. Since this is true for each edge-deleted subgraph, and Fisg(G) is full lattice determined, Fisg(G) ∼ = Fisg(H) Thus, by Theorem 4.1, G ∼ = H, and G is edge-reconstructible. □ Notice that the idempotents covered by idG in Iisg(G) are vertex-deleted subgraphs of G. It is also easy to see that vertex deleted subgraphs who are isomorphic will have isomorphic induced inverse semigroups. However, Proposition 4.2 does not provide the converse. Thus, we do not have a result as strong as Theorem 5.2. Hence, we propose the following question: Problem 1. Find necessary and sufficient properties for Iisg(G) so that G is vertex-reconstructible.
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Acknowledgments The authors are grateful to the referees whose helpful suggestions greatly improved this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
M.O. Albertson, K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1) (1996) R18. C.J. Ash, T.E. Hall, Inverse semigroups on graphs, Semigroup Forum 11 (1) (1975) 140–145. J.A. Bondy, R.L. Hemminger, Graph reconstruction –a survey, J. Graph Theory 1 (3) (1977) 227–268. J.A. Bondy, U.S.R. Murty, Graph theory. 2008, Grad. Texts in Math. (2008). T. Chih, The groupoid of induced subgraphs, 2016, under review. C.R. Gibbons, J.D. Laison, Fixing numbers of graphs and groups, Electron. J. Combin. 16 (1) (2009) R39. F. Harary, On the reconstruction of a graph from a collection of subgraphs, in: Theory of Graphs and Its Applications (Proc. Sympos. Smolenice, 1963), 1964, pp. 47–52. D.G. Jones, M.V. Lawson, Graph inverse semigroups: their characterization and completion, J. Algebra 409 (2014) 444–473. A.V. Kelarev, Graph Algebras and Automata, Marcel Dekker, 2003. A.V. Kelarev, On Cayley graphs of inverse semigroups, Semigroup Forum 72 (3) (2006) 411–418. A.V. Kelarev, J. Ryan, J. Yearwood, Cayley graphs as classifiers for data mining: the influence of asymmetries, Discrete Math. 309 (17) (2009) 5360–5369. P. Kelly, A congurence theorem for trees, Pacific J. Math. 7 (1957) 961–968. M.V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific Publishing Company, 1999. L. Lovász, A note on the line reconstruction problem, in: Classic Papers in Combinatorics, Springer, 1987, pp. 451–452. S.W. Margolis, J.C. Meakin, E-unitary inverse monoids and the Cayley graph of a group presentation, J. Pure Appl. Algebra 58 (1) (1989) 45–76. G.B. Preston, Representations of inverse Semigroups, J. Lond. Math. Soc. 29 (1954). D. Rees, On semigroups, Proc. Cambridge Philos. Soc. 36 (1940). N. Sieben, Cayley color graphs of inverse semigroups and groupoids, Czechoslovak Math. J. 58 (3) (2008) 683–692. S. Ulam, A Collection of Mathematical Problems, Wiley, 1960.