A note on Oct_{1}^{+}-free graphs and Oct_{2}^{+}-free graphs
aa r X i v : . [ m a t h . C O ] F e b A note on
Oct +1 -free graphs and Oct +2 -free graphs Wenyan Jia , Shuai Kou , Chengfu Qin , Weihua Yang ∗ Department of Mathematics, Taiyuan University of Technology,Taiyuan Shanxi-030024, China Department of Mathematics, Guangxi Teachers Education University, Nanning
Abstract:
Let
Oct +1 and Oct +2 be the planar and non-planar graphs that obtained fromthe Octahedron by 3-splitting a vertex respectively. For Oct +1 , we prove that a 4-connectedgraph is Oct +1 -free if and only if it is C , C k +1 ( k ≥
2) or it is obtained from C by repeatedly4-splitting vertices. We also show that a planar graph is Oct +1 -free if and only if it is constructedby repeatedly taking 0-, 1-, 2-sums starting from { K , K , K } ∪ K ∪ { Oct, L } , where K isthe set of graphs obtained by repeatedly taking the special 3-sums of K . For Oct +2 , we provethat a 4-connected graph is Oct +2 -free if and only if it is planar, C k +1 ( k ≥ L ( K , ) or it isobtained from C by repeatedly 4-splitting vertices.Keywords: Graph minor ; Split ; Octahedron; Let
Oct denote the Octahedron. We only consider simple graphs in this article. Let G and H be two graphs. H is called a minor of G denoted by H ≤ m G if it can be generated bydeleting or contracting edges from G . If G has no minor isomorphic to H , G is H -free. Assume ∗ Corresponding author. E-mail: [email protected], [email protected] is a vertex of a 3-connected graph G such that d ( v ) ≥
4. Given two sets
A, B ⊆ N G ( v ), where N G ( v ) is the set of vertices adjacent to v in G and A ∩ B = ∅ , min {| A | , | B |} ≥
2. We mean a3- split of v is the operation of first deleting v from G and adding two new adjacent vertices v ′ , v ′′ , then joining v ′ to vertices in A and v ′′ to vertices in B . It is clearly that a graph obtainedby 3-splitting a vertex of a 3-connected graph will also be 3-connected.For a given graph H , characterizing H -free graphs is a difficult topic in graph theory. Wefocus on 3-connected graph H in this paper. Tutte’s Wheel Theorem states that every 3-connected graph can be obtained from a wheel by repeatedly adding edges and 3-splittingvertices [12]. By this theorem, we can generate all 3-connected graphs.Let G , G be two disjoint graphs with more than k vertices. The 0-sum of G and G isthe disjoint union of G , G . The 1-sum of G , G is obtained by identifying one vertex of G with one vertex of G . The 2-sum of G , G is obtained by identifying one edge of G withone edge of G , and the common edge could be deleted after identification. The 3-sum of G , G is obtained by identifying one triangle of G with one triangle of G , and some of the threecommon edges could be deleted after identification.Next, we introduce some known results for H -free graphs where H is 3-connected and weorder the results according to the number of edges of H .Ding [3] characterized all H -free graphs for 3-connected H with at most 11 edges, including Oct \ e . Let ℵ denote the set of graphs obtained by 3-summing wheels and Prisms, and let K △ denote the graph obtained by 3-summing Prism and K . Theorem 1.1 ([3]) . Oct \ e -free graphs consists of graphs in ℵ and 3-connected minors of V , Cube , and K △ . For 3-connected graphs with 12 edges, V -free graphs, Cube -free graphs and
Oct -free graphsare characterized in [2, 6, 7].
Theorem 1.2 ([2]) . A graph is Oct-free if and only if it is constructed by 0-, 1-, 2- and 3-sumsstarting from graphs in { K , K , K , K } ∪ { C n − : n ≥ } ∪ { L ′ , L , L ′ , L ′′ , P } (see Figure1). There are 51 3-connected graphs with 13 edges, but only two related results. One is for L ′ , L , L ′ , L ′′ , P Oct + -free graphs, where Oct + denote the graph Oct + e [5]. It can be seen that Oct + is isomorphic to K with two parallel edges removed. Theorem 1.3 ([5]) . Every 4-connected graph that does not contain a minor isomorphic to
Oct + is either planar or the square of an odd cycle. In this paper, we consider the two graphs that obtained by 3-splitting a vertex of the Oc-tahedron. We denote the planar one by
Oct +1 and the non-planar one by Oct +2 (as shown inFigure 2). It can be seen that Oct +1 and Oct +2 are 3-connected and they both have 13 edges.Our purpose is to characterize 4-connected Oct +1 -free graphs and 4-connected Oct +2 -free graphs.For Oct +1 , we also characterize all planar Oct +1 -free graphs by characterizing 3-connected planar Oct +1 -free graphs. Figure 2:
Oct +1 , Oct +2 Let v be a vertex of a 4-connected graph G . A 4- split of v produces a new graph G ′ asfollows. Given two sets A, B ⊆ N G ( v ), where N G ( v ) is the set of vertices adjacent to v in G and min {| A | , | B |} ≥
3. Remove v from G and add two new adjacent vertices v ′ , v ′′ such that N G ′ ( v ′ ) = A ∪ { v ′′ } , N G ′ ( v ′′ ) = B ∪ { v ′ } . Clearly, G ′ is also 4-connected.The following are the main results of this paper. Theorem 1.4.
A 4-connected graph is
Oct +1 -free if and only if it is C , C k +1 ( k ≥ ) or it s obtained from C by repeatedly 4-splitting vertices. And C is the only 4-connected planar Oct +1 -free graph. Theorem 1.5.
A planar graph is
Oct +1 -free if and only if it is constructed by repeatedly taking0-, 1-, 2-sums starting from { K , K , K }∪ K ∪{ Oct, L } , where K is the set of graphs obtainedby repeatedly taking the special 3-sums of K . Figure 3: The special 3-sums of six K Theorem 1.6.
A 4-connected graph is
Oct +2 -free if and only if it is planar, C k +1 ( k ≥ ), L ( K , ) or it is obtained from C by repeatedly 4-splitting vertices. In this section, we introduce some definitions and known results we will use in section 3.A separation of G is an ordered pair of subgraphs ( H, K ) such that E ( H ) ∩ E ( K ) = ∅ , H ∪ K = G , and | E ( H ) | ≥ | V ( H ) ∩ V ( K ) | ≤ | E ( K ) | . A separation ( H, K ) is called a k - separation if | V ( H ) ∩ V ( K ) | = k . A cyclic separation is a separation ( H, K ) in which both H and K contain circuits. Suppose k is an integer greater than two. A graph G is cyclicallyk - connected if it is 2-connected, | E ( G ) | − | V ( G ) | + 1 ≥ k and there does not exist a cyclic k ′ -separation of G for k ′ ≤ k .A graph is cubic if it is 3-regular. The graph L ( H ) is called the line graph of G if V ( L ( G )) = E ( G ), and for any two vertices e , f in V ( L ( G )), e and f are adjacent vertices if and only ifthey are adjacent edges in G . Let C = { C n : n ≥ } and L = { L ( H ): H be the cubic cyclically4-connected graph } . A ( G , G n )- chain is a sequence of 4-connected graphs G , G , ..., G n andeach G i ( i < n ) has an edge e i such that G i /e i = G i +1 . here is a classical result of Martinov for 4-connected graphs, which is known as chaintheorem. Theorem 2.1 ([10]) . For every 4-connected graph G , there exists a sequence of 4-connectedgraphs G , G , ..., G n such that G = G , G n ∈ C ∪ L , and every G i ( i < n ) has an edge e i forwhich G i /e i = G i +1 . This result has been strengthened by Qin and Ding as follows.
Theorem 2.2 ([1]) . Let G be a 4-connected graph not in C ∪ L . If G is planar, then there existsa ( G, C ) -chain; if G is non-planar, then there exists a ( G, K ) -chain. Thus, any 4-connected graph that not in
C ∪ L can be generated by repeatedly 4-splittingvertices from C or C .There are some good properties for cubic cyclically 4-connected graphs and the line graphs.Adding a handle to G means the operation that first subdivide two nonadjacent edges e and e of G , then add a new edge connecting two new internal vertices. And a graph H is topologically contained in a graph G , if there exists a subgraph of G that is isomorphic to a subdivision of H . Lemma 2.3 ([9]) . The class of all cubic cyclically 4-connected graphs can be generated byrepeatedly adding handles starting from K , or the Cube . Lemma 2.4 ([4]) . If G is a cyclically 4-connected non-planar cubic graph, then either G = K , or G contains a subdivision of V . Lemma 2.5 ([8]) . If H is topologically contained in G , then L ( H ) ≤ m L ( G ) . The following are some results for 3-connected graphs.
Lemma 2.6 ([3]) . Let H be a 3-connected graph. Then a graph is H -free if and only if it isconstructed by repeatedly taking 0-, 1-, and 2-sums, starting from { K , K , K } ∪ { H -free graphs } . Lemma 2.7 ([11]) . Suppose a 3-connected graph H = W is a proper minor of a 3-connectedgraph G = W n . Then G has a minor J, which is obtained from H by either adding an edge or3-splitting a vertex. Proof of main results
Oct +1 -free graphs In this section, we characterize 4-connected
Oct +1 -free graphs and prove Theorem 1.4. Lemma 3.1.
If a 4-connected graph G ∈ C ∪ L and G is Oct +1 -free, then G is C or C k +1 ( k ≥ .Proof. Suppose G = L ( H ), where H is a cubic cyclically 4-connected graph. By Lemma 2.3, H can be generated by repeatedly adding handles starting from K , or the Cube . Thus, L ( H ) ≥ m L ( K , ) or L ( H ) ≥ m L ( Cube ) by lemma 2.5. Since L ( K , ) and L ( Cube ) both contain
Oct +1 asa minor (as shown in Figure 4 and Figure 5), G contains Oct +1 -minor too.contact v v v v v v v v v v v v v v v v v v v v Figure 4: L ( K , ) and its Oct +1 -minor contractthe heavy edges Figure 5: L ( Cube ) and its
Oct +1 -minor Thus we assume that G ∈ C . Suppose G = C k +1 ( k ≥ G is Oct +1 -free since C k +1 ( k ≥
2) is
Oct -free. For C k ( k ≥ C k +2 contains C k as a minor. Clearly, C is Oct +1 -freesince it only has six vertices. And it is easy to verify that C contains Oct +1 -minor, thus all C k contains Oct +1 -minor for k ≥ heorem 3.2. A 4-connected planar graph is
Oct +1 -free if and only if it is C .Proof. The sufficiency clearly holds. To prove the necessity, assume that G is a 4-connectedplanar Oct +1 -free graph. If G ∈ C ∪ L , by Lemma 3.1 G = C . If G is not in C ∪ L , byTheorem 2.2 there exists a (
G, C )-chain.Let { v , v , ..., v } be vertices of C (shown in Figure 6). By symmetry, we 4-split v and firstconsider the minimal case | A | = | B | = 3, where A , B belongs to N ( v ), A ∪ B = N ( v ). Suppose v ′ , v ′′ are two new vertices obtained by 4-splitting v , and N ( v ′ ) = A ∪ { v ′′ } , N ( v ′′ ) = B ∪ { v ′ } .Since the four neighbors of v in C construct a 4-cycle, three vertices in A are pairwise adjacent.By symmetry, we assume that A = { v , v , v } , then v must be adjacent to v ′′ . Therefore, B must be one of the following sets: { v , v , v } , { v , v , v } , { v , v , v } . In all cases, the newgraph G ′ generated by 4-splitting v from C with | A | = | B | = 3 contains Oct +1 . Clearly, allother 4-splits of C contain G ′ as a minor, thus contain Oct +1 . Hence, G is C . v v v v v v v ’ v ” v v v v v v ’ v ” v v v v v v ’ v ” v v v v v Figure 6: C and three graphs obtained by minimal 4-splitting v Proof of Theorem 1.4
The result follows from Theorem 2.2, Lemma 3.1 and Theorem 3.2.
Oct +1 -free graphs In this section, we characterize all planar
Oct +1 -free graphs and prove Theorem 1.5. We firstestablish some lemmas. Lemma 3.3. If G , G are k -connected for k = 0 , , , and at least one of them is non-planar,then the k -sum of G and G is non-planar. roof. It clearly holds for k = 0 ,
1. Without loss of generality, we suppose G is non-planar, G is planar. Then G contains a subdivision of K , or K , we denote it by Γ. When k = 2, let G be the 2-sum of G and G , let e = uv be the common edge. Suppose G is planar, then e is contained in the subdivision Γ and e is deleted after identification. Since G is 2-connected,there exists an ( u , v )-path P different from e . Then G ∪ P contains a subdivision of K , or K ,a contradiction.When k = 3, let G be the graph obtained by 3-summing G and G over a common triangle v v v v . Suppose G is planar and v v , v v , v v are all deleted after identification. Since G is non-planar, some edges in { v v , v v , v v } are contained in the subdivision Γ. If Γ isa subdivision of K , , since there is no triangle in Γ, at most two edges of Γ are containedin { v v , v v , v v } , say v v , v v . Since G is 3-connected, there exists a vertex v differentfrom v , v , v and three internally-disjoint ( v, v )-path, ( v, v )-path, ( v, v )-path in G . Bycontracting ( v, v )-path to v , we can obtain a ( v , v )-path P and a ( v , v )-path P . ThenΓ \ v v \ v v ∪ P ∪ P forms a subdivision of K , again. This contradicts to the planarity of G . Thus we assume that Γ is a subdivision of K . As shown in Figure 7, the 3-sum G ′ of K and K is non-planar. Since every 3-connected graph contains W = K as a minor, G mustcontain G ′ as a minor. A contradiction. v v v Figure 7: The 3-sum of K and K It is sufficient to consider the k -sums ( k = 0 , , ,
3) of planar graphs to characterize allplanar
Oct +1 -free graphs. Lemma 3.4. If G , G are both planar, then the k -sum G ( k = 0 , , of G and G is planar.Proof. It clearly holds for k = 0 ,
1. When k = 2, let e be the common edge. Since G i ( i = 1 , s planar, there exists a planar embedding H i of G i such that the outer face e f i of H i is incidentwith e . Thus a planar embedding of G can be obtained by embedding H in e f of H .Recall that the 3-sum of G , G is obtained by identifying one triangle of G with onetriangle of G , and some of the three common edges could be deleted after identification. Next,we define the special sum of two graphs. A triangle abca of G is called a separating triangle if the graph obtained from G by deleting vertices a , b , c is disconnected. Otherwise, we call thetriangle abca non - separating triangle . The special sum of G and G is obtained by taking3-sum of them over a non-separating triangle of both G and G . Lemma 3.5.
Let G , G be two 3-connected planar graphs with triangles and let G be a 3-sumof them. If G is obtained by taking special 3-sum of them, then G is planar. Otherwise, G isnon-planar.Proof. Suppose G is obtained by 3-summing G , G over an non-separating triangle C of both G and G . Let f be the face of G that bounded by C . Then there exists a planar embedding H of G such that the outer face ˜ f of H has the same boundary as f . Thus a planar embeddingof G can be obtained by embedding G in ˜ f .Next, we suppose that G is obtained by 3-summing G and G over a separating triangle C = abca of G or G , say G . Since G is planar, there exist two vertices u , u such that u is in int C and u is in ext C . Let u be a vertex of G that different from { a, b, c } . Since both G and G are 3-connected, there exists a { u i , a } -path P i , a { u i , b } -path P i and a { u i , c } -path P i in G such that P i , P i and P i are internally-disjoint (Shown in Figure 8). It can be seenthat P ∪ P ∪ P ∪ P ∪ P ∪ P ∪ P ∪ P ∪ P forms a subdivision of K , . Thus, G isnon-planar. a b cu u u Figure 8: A subdivision of K , e next prove Theorem 1.5. Proof of Theorem 1.5
We first characterize all 3-connected planar
Oct +1 -free graphs. Suppose a graph G is 3-connected planar Oct +1 -free graph. Two cases now arise, depending on whether G has an Oct-minor. Case 1. G contains an Oct -minor.It is clearly that G = Oct is 3-connected planar
Oct +1 -free graph. Thus we assume that G = Oct . By Lemma 2.7, G has a minor J , which is obtained from Oct by either adding an edgeor 3-splitting a vertex. Adding any edge to
Oct results in a non-planar graph. And there aretwo graphs obtained by 3-splitting a vertex of
Oct , one is
Oct +1 , another is non-planar. Hence,in this case, Oct is the only 3-connected planar
Oct +1 -free graph. Case 2. G is Oct -free.Since G is 3-connected, G is constructed by taking 3-sums starting from graphs in { K } ∪{ C n − : n ≥ } ∪ { L ′ , L , L ′ , L ′′ , P } by Theorem 1.2. By Lemma 3.3, Lemma 3.5 and theplanarity of G , G is L or G is the special 3-sums of K . Thus G belongs to { L } ∪ K in thiscase.It follows from Case 1 and Case 2 that G belongs to { Oct, L } ∪ K . Then Theorem 1.5follows from Lemma 2.6, Lemma 3.3 and Lemma 3.4. Oct +2 -free graphs In this section, we characterize 4-connected
Oct +2 -free graphs and prove Theorem 1.6. Lemma 3.6.
Graphs in C are all 4-connected Oct +2 -free graphs.Proof. It clearly holds since C k ( k ≥
3) is planar and C k +1 ( k ≥
2) is
Oct -free.
Lemma 3.7.
If a 4-connected graph G ∈ L and G is Oct +2 -free, then G is planar or G = L ( K , ) .Proof. Suppose G = L ( H ), where H is a cubic cyclically 4-connected graph. If G is planar,then G is Oct +2 -free since Oct +2 is non-planar. When G is non-planar, H is non-planar too. By emma 2.4, H = K , or H contains a subdivision of V . Case 1. H = K , .As shown in Figure 9, G = L ( K , ) has 9 vertices { v , v , ..., v } and 18 edges. If Oct +2 is aminor of G , the minor can be obtained from G by contracting two edges and then deleting someedges. Without loss of generality, we first contract v v to v and denote the resulting graphby G , means it has 8 vertices and 16 edges. By symmetry, we next contract one of the edgesin { v v , v v , v v , v v , v v , v v , v v , v v , v v , v v } . We verify every case in order and upto isomorphism there are six resulting graphs, we denote them by G a , G a , G b , G b , G , G c respectively. Since they are all Oct +2 -free graphs, G = L ( K , ) is Oct +2 -free too. v v vv v vv v v v v vv vv v v Figure 9: L ( K , ), G , G a , G a , G b , G b , G , G c contract theheavy edges delete thedashed edges Figure 10: L ( V ) and its Oct +2 -minor Case 2. H contains a subdivision of V .By Lemma 2.5, G contains L ( V ) as a minor. Since L ( V ) contains a Oct +2 -minor as shownin Figure 10, G contains Oct +2 as a minor. roof of Theorem 1.6 Suppose G is a 4-connected graph that is not in C ∪ L . If G is planar, then G is Oct +2 -freeclearly. If G is non-planar, by Theorem 2.2, there is a ( G, K )-chain. That is G can be generatedby repeatedly splitting vertices of C .Then Theorem 1.6 follows from Lemma 3.6 and Lemma 3.7. References [1] C. Qin, and G. Ding, A chain theorem for 4-connected graphs, Journal of CombinatorialTheory Series B. 134 (2019) 341–349.[2] G. Ding, A characterization of graphs with no octahedron minor, Journal of Graph Theory74(2) (2013) 143–162.[3] G. Ding, and C. Liu, Excluding a small minor, Discrete Applied Mathematics 161(3) (2013)355–368.[4] G. Ding, C. Lewchalermvongs, and J. Maharry, Graphs with no ¯ P -minor, The ElectronicJournal of Combinatorics 23(2) (2016) P2.12.[5] J. Maharry, An excluded minor theorem for the octhedron plus an edge, Journal of GraphTheory 57(2) (2008) 124–130.[6] J. Maharry, and N. Robertson, The structure of graphs not topologically containing theWagner graph, Journal of Combinatorial Theory Series B. 121 (2016) 398–420[7] J. Maharry, A characterization of graphs with no cube minor, Journal of CombinatorialTheory Series B. 80 (2008) 179–201[8] J. Maharry, An excluded minor theorem for the octahedron, Journal of Graph Theory 31(2)(1999) 95–100.[9] N.C. Wormald, Classifying k-connected cubic graphs, Lect Note Math. 748 (1979) 199–206.
10] N. Martinov, Uncontractible 4-connected graphs, Journal of Graph Theory 6 (1982) 343–344.[11] P. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory SeriesB. 28 (1980) 305–359.[12] W.T. Tutte, A theory of 3-connected graphs, Proceedings of the Koninklijke NederlandseAkademie van Wetenschappen Series A. 64 (1961) 441–455.10] N. Martinov, Uncontractible 4-connected graphs, Journal of Graph Theory 6 (1982) 343–344.[11] P. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory SeriesB. 28 (1980) 305–359.[12] W.T. Tutte, A theory of 3-connected graphs, Proceedings of the Koninklijke NederlandseAkademie van Wetenschappen Series A. 64 (1961) 441–455.