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Research article

The k-subconnectedness of planar graphs

  • Received: 13 November 2020 Accepted: 01 March 2021 Published: 26 March 2021
  • MSC : 05C40, 05C85

  • A graph G with at least 2k vertices is called k-subconnected if, for any 2k vertices x1,x2,,x2k in G, there are k independent paths joining the 2k vertices in pairs in G. In this paper, we prove that a k-connected planar graph with at least 2k vertices is k-subconnected for k=1,2; a 4-connected planar graph is k-subconnected for each k such that 1kν/2, where v is the number of vertices of G; and a 3-connected planar graph G with at least 2k vertices is k-subconnected for k=4,5,6. The bounds of k-subconnectedness are sharp.

    Citation: Zongrong Qin, Dingjun Lou. The k-subconnectedness of planar graphs[J]. AIMS Mathematics, 2021, 6(6): 5762-5771. doi: 10.3934/math.2021340

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  • A graph G with at least 2k vertices is called k-subconnected if, for any 2k vertices x1,x2,,x2k in G, there are k independent paths joining the 2k vertices in pairs in G. In this paper, we prove that a k-connected planar graph with at least 2k vertices is k-subconnected for k=1,2; a 4-connected planar graph is k-subconnected for each k such that 1kν/2, where v is the number of vertices of G; and a 3-connected planar graph G with at least 2k vertices is k-subconnected for k=4,5,6. The bounds of k-subconnectedness are sharp.



    Connectivity is an important property of graphs. It has been extensively studied (see [1]). A graph G=(V,E) is called k-connected (k1) (k-edge-connected) if, for any subset SV(G) (SE(G)) with |S|<k, GS is connected. The connectivity κ(G) (edge connectivity λ(G)) is the order (size) of minimum cutset (edge cutset) SV(G) (SE(G)). When G is a complete graph Kn, we define that κ(G)=n1.

    In recent years, conditional connectivities attract researchers' attention. For example, Peroche [2] studied several sorts of connectivities, including cyclic edge (vertex) connectivity, and their relations. A cyclic edge (vertex) cutset S of G is an edge (vertex) cutset whose deletion disconnects G such that at least two of the components of GS contain a cycle respectively. The cyclic edge (vertex) connectivity, denoted by cλ(G) (cκ(G)), is the cardinality of a minimum cyclic edge (vertex) cutset of G. Dvoŕǎk, Kára, Král and Pangrác [3] obtained the first efficient algorithm to determine the cyclic edge connectivity of cubic graphs. Lou and Wang [4] obtained the first efficient algorithm to determine the cyclic edge connectivity for k-regular graphs. Then Lou and Liang [5] improved the algorithm to have time complexity O(k9V6). Lou [6] also obtained a square time algorithm to determine the cyclic edge connectivity of planar graphs. In [7], Liang, Lou and Zhang obtained the first efficient algorithm to determine the cyclic vertex connectivity of cubic graphs. Liang and Lou [8] also showed that there is an efficient algorithm to determine the cyclic vertex connectivity for k-regular graphs with any fixed k.

    Another related concept is linkage. Let G be a graph with at least 2k vertices. If, for any 2k vertices u1,u2,,uk,v1,v2,,vk, there are k disjoint paths Pi from ui to vi(i=1,2,,k) in G, then G is called k-linked. Thomassen [9] mentioned that a necessary condition for G to be k-linked is that G is (2k1)-connected. But this condition is not sufficient unless k=1. He also gave a complete characterization of 2-linked graphs. Bollobás and Thomason [10] proved that if κ(G)22k, then G is k-linked. Kawarabayashi, Kostochka and Yu [11] proved that every 2k-connected graph with average degree at least 12k is k-linked.

    In [12], Qin, Lou, Zhu and Liang introduced the new concept of k-subconnected graphs. Let G be a graph with at least 2k vertices. If, for any 2k vertices v1,v2,,v2k in G, there are k vertex-disjoint paths joining v1,v2,,v2k in pairs, then G is called k-subconnected. If G is k-subconnected and ν(G)3k1, then G is called a properly k-subconnected graph. In [12], Qin et al. showed that a properly k-subconnected graph is also a properly (k1)-subconnected graph. But only when ν(G)3k1, that G is k-subconnected implies that G is (k1)-subconnected. Qin et al. [12] also gave a sufficient condition for a graph to be k-subconnected and a necessary and sufficient condition for a graph to be a properly k-subconnected graph (see Lemmas 1 and 2 and Corollary 3 in this paper).

    If G has at least 2k vertices, that G is k-linked implies that G is k-connected, while that G is k-connected implies that G is k-subconnected (see Lemma 6 in this paper). Also in a k-connected graph G, deleting arbitrarily some edges from G, the resulting graph H is still k-subconnected. So a graph H to be k-subconnected is a spanning substructure of a k-connected graph G. To study k-subconnected graphs may help to know more properties in the structure of k-connected graphs. Notice that a k-connected graph may have a spanning substructure to be m-subconnected for m>k.

    K-subconnected graphs have some background in matching theory. The proof of the necessary and sufficient condition [12] for properly k-subconnected graphs uses similar technique to matching theory.

    Let S be a subset of V(G) of a graph G. We denote by G[S] the induced subetaaph of G on S. We also denote by ω(G) the number of components of G. We also use ν(G) and ε(G) to denote |V(G)| and |E(G)|. If G is a planar graph, we denote by ϕ(G) the number of faces in the planar embedding of G. Let H be a graph. A subdivision of H is a graph H obtained by replacing some edges by paths respectively in H. For other terminology and notation not defined in this paper, the reader is referred to [13].

    In this section, we shall present some known results and some straightforward corollaries of the known results which will be used in the proof of our main theorems.

    Lemma 1 (Theorem 1 of [12]). Let G be a connected graph with at least 2k vertices. Then G is k-subconnected if, for any cutset SV(G) with |S|k1, ω(GS)|S|+1.

    Lemma 2 (Theorem 2 of [12]). Let G be a connected graph with at least 3k1 vertices. If G is a properly k-subconnected graph, then, for any cutset SV(G) with |S|k1, ω(GS)|S|+1.

    Only when v3k1, that G is k-subconnected implies that G is (k-1)-subconnected. Here is an counterexample. Let S=Kk2 be a complete graph of k2 vertices, let H be k copies of K2, and let G be a graph with V(G)=V(S)V(H) and E(G)=E(S)E(H){uv|uV(S),vV(H)}. Then v(G)=3k2, and G is not (k-1)-subconnected since we can choose 2(k-1) vertices by taking one vertex from each copy of K2 in H and taking all vertices of S, then these 2(k-1) vertices cannot be joined by k-1 independent paths in pairs. But G is k-subconnected since when we take any 2k vertices from G, some pairs of vertices will be taken from several same K2 's in H, and then the 2k vertices can be joined by k independent paths in pairs.

    Corollary 3 (Theorem 3 of [12]). Let G be a connected graph with at least 3k1 vertices. Then G is a properly k-subconnected graph if and only if, for any cutset SV(G) with |S|k1, ω(GS)|S|+1.

    Lemma 4 ([14]). Every 4-connected planar graph is Hamiltonian.

    Lemma 5. If a graph G has a Hamilton path, then G is k-subconnected for each k such that 1kν(G)/2.

    Proof. Let P be a Hamilton path in G. Let vi, i=1,2,,2k, be any 2k vertices in V(G). Without loss of generality, assume that v1,v2,,v2k appear on P in turn. Then there are k paths Pi on P from v2i1 to v2i, i=1,2,,k, respectively. So G is k-subconnected.

    Lemma 6. A k-connected graph G with at least 2k vertices is k-subconnected.

    Proof. Let G be a k-connected graph with at least 2k vertices. Then G does not have a cutset SV(G) with |S|k1, so the statement that, for any cutset SV(G) with |S|k1, ω(GS)|S|+1 is true. By Lemma 1, G is k-subconnected.

    In this section, we shall show the k-subconnectedness of planar graphs with different connectivities, and show the bounds of k-subconnectedness are sharp.

    Corollary 7. A 1-connected planar graph G with at least 2 vertices is 1-subconnected.

    Proof. By Lemma 6, the result follows.

    Corollary 8. A 2-connected planar graph G with at least 4 vertices is 2-subconnected.

    Proof. By Lemma 6, the result follows.

    Theorem 9. A 4-connected planar graph G is k-subconnected for each k such that 1kν(G)/2.

    Proof. By Lemma 4, G has a Hamilton cycle C, and then has a Hamilton path P. By Lemma 5, the result follows.

    Theorem 10. A 3-connected planar graph G with at least 2k vertices is k-subconnected for k=4,5,6.

    Proof. Suppose that G is a 3-connected planar graph with at least 2k vertices which is not k-subconnected. By Lemma 1, there is a cutset SV(G) with |S|k1 such that ω(GS)|S|+2. Since G is 3-connected, there is no cutset with less than 3 vertices and so |S|3. On the other hand, k=4,5,6, so |S|5. Thus let us consider three cases.

    In the first case, |S|=3. By our assumption, ω(GS)|S|+2, let C1,C2,,C5 be different components of GS, and S={x1,x2,x3}. Since G is 3-connected, every Ci is adjacent to each xj (1i5,1j3). Contract every Ci to a vertex Ci (i=1,2,,5) to obtain a planar graph G as G is planar. Then G[{x1,x2,x3}{C1,C2,C3}] contains a K3,3, which contradicts the fact that G is a planar graph.

    In the second case, |S|=4. By our assumption, ω(GS)|S|+2, let C1,C2,,C6 be different components of GS and S={x1,x2,x3,x4}. Contract every Ci to a vertex Ci (i=1,2,,6) to obtain a planar graph G as G is planar. Since G is 3-connected, each Ci is adjacent to at least 3 vertices in S (1i6). (In the whole proof, we shall consider that Ci is adjacent to only 3 vertices in S, and we shall neglect other vertices in S which are possibly adjacent to Ci). Since the number of 3-vertex-combinations in S is C(4,3)=4, but C1,C2,,C6 have 6 vertices, by the Pigeonhole Principle, there are two vertices in {C1,C2,,C6} which are adjacent to the same three vertices in S. Without loss of generality, assume that C1 and C2 are both adjacent to x1,x2,x3. If there is another Ci (3i6) adjacent to x1,x2,x3, say C3, then G[{x1,x2,x3}{C1,C2,C3}] contains a K3,3, which contradicts the fact that G is planar (which also contradicts the assumption that G is a planar graph because G has a subetaaph which can be contracted to a K3,3). So Ci cannot be adjacent to x1,x2,x3 at the same time (i=3,4,5,6).

    Suppose C3 is adjacent to x2,x3,x4. If one of C4,C5,C6 is adjacent to both x1 and x4, say C4, then C3 is connected to x1 by path C3x4C4x1, so G[{x1,x2,x3,x4}{C1,C2,C3,C4}] contains a subdivision of K3,3, contradicting the fact that G is planar. Since C4,C5,C6 are all not adjacent to x1,x2,x3 at the same time, they are all adjacent to x4. But each of them cannot be adjacent to both x1 and x4. So they are all not adjacent to x1. Hence C4,C5,C6 are all adjacent to x2,x3,x4 at the same time. Then G[{x2,x3,x4}{C4,C5,C6}] contains a K3,3, contradicting the fact that G is a planar graph.

    The cases that C3 is adjacent to x1,x3,x4 or x1,x2,x4 are similar.

    In the third case, |S|=5. By our assumption, ω(GS)|S|+2, let C1,C2,,C7 be different components of GS and S={x1,x2,,x5}. Since G is planar, contracting Ci to a vertex Ci (i=1,2,,7), we obtain a planar graph G. Also since G is 3-connected, every Ci is adjacent to at least 3 vertices in S (1i7).

    Case 1. There are two of Ci (i=1,2,,7) adjacent to the same three vertices in S. Without loss of generality, assume that C1 and C2 are both adjacent to x1,x2,x3 at the same time.

    If there is another vertex Ci (3i7) adjacent to x1,x2,x3 at the same time, say C3. Then G[{x1,x2,x3}{C1,C2,C3}] contains a K3,3, contradicting the fact that G is planar. So C3 cannot be adjacent to x1,x2,x3 at the same time. Without loss of generality, we have two subcases.

    Suppose C3 is only adjacent to two vertices in {x1,x2,x3}, say x2 and x3. Then C3 must be adjacent to one of x4 and x5 as G is 3-connected. Without loss of generality, assume that C3 is also adjacent to x4. If one of Ci (i=4,5,6,7) is adjacent to both x1 and x4, say C4. Then C3 is connected to x1 by path C3x4C4x1, hence G[{x1,x2,x3,x4}{C1,C2,C3,C4}] contains a subdivision of K3,3, contradicting the fact that G is planar. So none of Ci (i=4,5,6,7) is adjacent to both x1 and x4.

    Case (1.1). Suppose that C4 is adjacent to three vertices in {x1,x2,x3,x4} .

    If C4 is adjacent to x1,x2,x3 at the same time, then the case is similar to that C3 is adjacent to x1,x2,x3 at the same time, and we have a contradiction. So C4 is not adjacent to x1,x2,x3 at the same time. If C4 is adjacent to x1, since C4 is adjacent to three vertices in {x1,x2,x3,x4} but not x1,x2,x3, so C4 is adjacent to x1 and x4, by the argument above, we have a contradiction. Hence C4 can be adjacent only to x2,x3,x4.

    Now x1 and x4 are symmetric, while x2 and x3 are symmetric in G[{x1,x2,x3,x4}{C1,C2,C3,C4}]. Then Ci(i=5,6,7) must be adjacent to x5, we have two cases as follows.

    Notice that now there are not i and j, 5ij7, such that Ci is adjacent to x1 and Cj is adjacent to x4, otherwise C3 is connected to x1 by path C3x4Cjx5Cix1, then G[{x1,x2,x3,x4,x5}{C1,C2,C3,Ci,Cj}] contains a subdivision of K3,3, contrary to the fact that G is planar.

    Suppose C5 is adjacent to x3,x4,x5. If C6 is also adjacent to x3,x4,x5, then C7 cannot be adjacent to x3,x4,x5, otherwise G[{x3,x4,x5}{C5,C6,C7}] contains a K3,3, a contradiction. So suppose C7 is adjacent to x2,x4,x5, then C7 is connected to x3 by path C7x2C2x3, and then G[{x2,x3,x4,x5}{C2,C5,C6,C7}] contains a subdivision of K3,3, a contradiction. So suppose C7 is adjacent to x2,x3,x5. But C7 is connected to x4 by path C7x2C3x4, then G[{x2,x3,x4,x5}{C3,C5,C6,C7}] contains a subdivision of K3,3, a contradiction again. If C6 is adjacent to x2,x3,x5, suppose C7 is adjacent to x3,x4,x5, then this case is similar to that C6 is adjacent to x3,x4,x5 and C7 is adjacent to x2,x3,x5. Then suppose C7 is adjacent to x2,x3,x5. Now C3 is connected to x5 by path C3x4C5x5, so G[{x2,x3,x4,x5}{C3,C5,C6,C7}] contains a subdivision of K3,3, a contradiction. Now suppose C7 is adjacent to x2,x4,x5. Then C6 is connected to x4 by path C6x5C7x4, hence G[{x2,x3,x4,x5}{C3,C4,C6,C7}] contains a subdivision of K3,3, a contradiction. The remaining case is that C6 is adjacent to x2,x4,x5. Now the cases that C7 is adjacent to x2,x4,x5 and that C7 is adjacent to x3,x4,x5 are symmetric, we only discuss the former. Then C5 is connected to x2 by path C5x3C4x2, so G[{x2,x3,x4,x5}{C4,C5,C6,C7}] contains a subdivision of K3,3, a contradiction. The remaining case is that C7 is adjacent to x2,x3,x5. Now C7 is connected to x4 by path C7x5C6x4. Hence G[{x2,x3,x4,x5}{C3,C4,C6,C7}] contains a subdivision of K3,3, a contradiction.

    Suppose C5 is adjacent to x2,x3,x5. If C6 and C7 are both adjacent to x2,x3,x5, then G[{x2,x3,x5}{C5,C6,C7}] contains a K3,3, contradicting the fact that G is planar. So one of C5, C6, C7 is adjacent to x2,x3,x5, the other two are adjacent to x3,x4,x5; or one of C5, C6, C7 is adjacent to x3,x4,x5, the other two are adjacent to x2,x3,x5; or C5 is adjacent to x2,x3,x5, C6 is adjacent to x3,x4,x5 and C7 is adjacent to x2,x4,x5. These three cases are symmetric to cases discussed above. (Notice that the roles of C5, C6 and C7 are symmetric.)

    Case (1.2). Now suppose {x2,x3,x4} - N(C4).

    Notice that {x2,x3,x4} - N(Ci) for 5i7, otherwise the Ci (5i7) is similar to C4 as discussed above, and the other three in {C4,C5,C6,C7} are similar to C5,C6,C7, by the same argument as above, we obtain a contradiction. Also since C4,C5,C6,C7 each cannot be adjacent to both x1 and x4, all of C4,C5,C6,C7 must be adjacent to x5. Now x1 and x4 are not symmetric but x2 and x3 are symmetric in G[{x1,x2,x3,x4}{C1,C2,C3}].

    First, suppose C4 is adjacent to x4 and x3 besides x5. Then suppose C5 is adjacent to x4 and x3. If C6 is also adjacent to x4 and x3, then G[{x3,x4,x5}{C4,C5,C6}] contains K3,3, a contradiction. Then suppose C6 is adjacent to x4,x2, now C6 is connected to x3 by path C6x2C2x3, and G[{x2,x3,x4,x5}{C2,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. Now suppose C6 is adjacent to x4,x1, then C6 is connected to x3 by path C6x1C2x3, and G[{x1,x3,x4,x5}{C2,C4,C5,C6}] contains a subdivision of K3,3 a contradiction. Then suppose C6 is adjacent to x3,x2. Now C6 is connected to x4 by path C6x2C3x4, so G[{x2,x3,x4,x5}{C3,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. Next we suppose C6 is adjacent to x3,x1, then C6 is connected to x2 by path C6x5C4x4C3x2, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction. Then suppose C6 is adjacent to x2,x1, now C6 is connected to x3 by path C6x5C4x3, and G[{x1,x2,x3,x5}{C1,C2,C4,C6}] contains a subdivision of K3,3, a contradiction. Now we suppose C5 is adjacent to x4,x2. By the symmetry of the roles of C4,C5,C6,C7, we only consider the cases of C6 as follows. Suppose C6 is adjacent to x4,x2. Then C4 is connected to x2 by path C4x3C3x2, and G[{x2,x3,x4,x5}{C3,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. Then suppose C6 is adjacent to x4,x1. Now C3 is connected to x1 by path C3x4C6x1, and G[{x1,x2,x3,x4}{C1,C2,C3,C6}] contains a subdivision of K3,3, a contradiction. Now suppose C6 is adjacent to x3,x2. Then we consider C7. By the symmetry of the roles of C7 and C6, we only consider the cases that C7 is adjacent to {x3,x2},{x3,x1},{x2,x1} respectively. If C7 is adjacent to x3,x2, then C5 is connected to x3 by path C5x4C4x3, and G[{x2,x3,x4,x5}{C4,C5,C6,C7}] contains a subdivision of K3,3, a contradiction. If C7 is adjacent to x3,x1, then C3 is connected to x1 by path C3x4C5x5C7x1, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C5,C7}] contains a subdivision of K3,3, a contradiction. If C7 is adjacent to x2,x1, then C7 is connected to x3 by path C7x5C4x3, and G[{x1,x2,x3,x5}{C1,C2,C4,C7}] contains a subdivision of K3,3, a contradiction. Now suppose C6 is adjacent to x3,x1. Then C3 is connected to x1 by path C3x4C4x5C6x1, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction. Now we suppose C6 is adjacent to x2,x1. Then C3 is connected to x1 by path C3x4C4x5C6x1, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction. Now suppose C5 is adjacent to x4,x1. Then C3 is connected to x1 by path C3x4C4x5C5x1, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C5}] contains a subdivision of K3,3, a contradiction. Then we suppose C5 is adjacent to x3,x2. By symmetry of C5 and C6, we only need to consider the following cases. Suppose C6 is adjacent to x3,x2. Then C3 is connected to x5 by path C3x4C4x5, and G[{x2,x3,x4,x5}{C3,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. Now suppose C6 is adjacent to x3,x1, then C3 is connected to x1 by path C3x4C4x5C6x1, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction. Then suppose C6 is adjacent to x2,x1, by the same argument as last case, we also obtain a contradiction. Now we suppose C5 is adjacent to x3,x1. Then C5 is connected to x2 by path C5x5C4x4C3x2, and G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C5}] contains a subdivision of K3,3, a contradiction. So suppose C5 is adjacent to x2,x1. Then C5 is connected to x3 by path C5x5C4x3, and G[{x1,x2,x3,x5}{C1,C2,C4,C5}] contains a subdivision of K3,3, a contradiction. Since the case that C4 is adjacent to x4,x2 is symmetric to the case that C4 is adjacent to x4,x3, we do not discuss this case. The case that C4 is adjacent to x4,x1 is excluded by the discussion before Case (1.1). So we suppose C4 is adjacent to x3,x2 now. By symmetry, we only need to consider the following cases. Suppose C5 is adjacent to x3,x2, then suppose C6 is adjacent to x3,x2. Then G[{x2,x3,x5}{C4,C5,C6}] contains a K3,3, a contradiction. Then suppose C6 is adjacent to x3,x1. Now C6 is connected to x2 by path C6x1C1x2, and G[{x1,x2,x3,x5}{C1,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. So suppose that C6 is adjacent to x2,x1, then C6 is connected to x3 by path C6x1C1x3, and G[{x1,x2,x3,x5}{C1,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. Now suppose C5 is adjacent to x3,x1, then suppose C6 is adjacent to x3,x1. Then C4 is connected to x1 by path C4x2C1x1, and G[{x1,x2,x3,x5}{C1,C4,C5,C6}] contains a subdivision of K3,3, a contradiction. So suppose C6 is adjacent to x2,x1. Now C6 is connected to x3 by path C6x5C4x3, and G[{x1,x2,x3,x5}{C1,C2,C4,C6}] contains a subdivision of K3,3, a contradiction. Then suppose C5 is adjacent to x2,x1, and C6 can be adjacent only to x2,x1, by the same argument as last case, we can obtain a contradiction. Now suppose C4 is adjacent to x3,x1, then suppose C5 and C6 are both adjacent to x3,x1. But G[{x1,x3,x5}{C4,C5,C6}] contains K3,3, a contradiction. So we suppose C6 is adjacent to x2,x1 subject to the above assumption of C4 and C5. Now C6 is connected to x3 by path C6x5C4x3, and G[{x1,x2,x3,x5}{C1,C2,C4,C6}] contains a subdivision of K3,3, a contradiction. Then suppose C5 is adjacent to x2,x1. Substituting C5 for C6 in the discussion of last case, we can obtain a contradiction. The remaining case is that C4,C5,C6 are all adjacent to x2,x1. Then G[{x1,x2,x5}{C4,C5,C6}] contains K3,3, a contradiction.

    Now we come back to the discussion of the first paragraph in Case 1. We have the second subcase as follows.

    Suppose C3 is adjacent to only one vertex in {x1,x2,x3}. By the symmetry of the roles of C3,C4,,C7, we can assume that each of C3,C4,,C7 is adjacent to only one vertex in {x1,x2,x3}. So all of C3,C4,,C7 are adjacent to both x4 and x5, and one of x1,x2,x3. But x1,x2,x3 have only 3 vertices and C3,C4,,C7 have 5 vertices, by the Pigeonhole Principle, there are Ci and Cj (3ij7) adjacent to x4,x5 and the same xr in {x1,x2,x3}, and there is a Ck (ki,j and 3k7) such that Ck is adjacent to x4,x5. We use Ci and Cj to replace C1 and C2, and Ck to replace C3, then Case 1 still happens. The proof is the same as before.

    Now suppose that Case 1 does not happen. Then, for any two vertices Ci and Cj (1i<j7), |N(Ci)N(Cj)|2.

    Case 2. Suppose there are two vertices Ci and Cj (1i<j7) such that |N(Ci)N(Cj)|=2.

    Without loss of generality, assume that N(C1)={x1,x2,x3}, N(C2)={x2,x3,x4} such that |N(C1)N(C2)|=2.

    Case (2.1). C1,C2,C3,C4 are adjacent to only vertices in {x1,x2,x3,x4}.

    Since G is 3-connected and Case 1 does not happen, then N(C3)={x1,x2,x4} and N(C4)={x1,x3,x4}. In G[{x1,x2,x3,x4}{C1,C2,C3,C4}], any two of x1,x2,x3,x4 are symmetric, and any two of C1,C2,C3,C4 are symmetric. Since Case 1 does not happen, N(Cj) is not contained in {x1,x2,x3,x4} (j=5,6,7). So Cj must be adjacent to x5 (j=5,6,7). By the symmetry of any two of x1,x2,x3,x4, we can assume that C5 is adjacent to x3,x4. Also by the assumption of Case 2, C6 is adjacent to x1 and x2, or adjacent to x2 and x3. In the first case, {x1,x2} and {x3,x4} are symmetric, then C1 is adjacent to x1,x2,x3, C2 is adjacent to x2,x3, and C2 is connected to x1 by path C2x4C3x1, C6 is adjacent to x1,x2, and C6 is connected to x3 by path C6x5C5x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C5,C6}] contains a subdivision of K3,3, a contradiction. In the second case, C2 is adjacent to x2,x3,x4, C4 is adjacent to x3,x4, and is connected to x2 by path C4x1C1x2, C5 is adjacent to x3,x4, and is connected to x2 by path C5x5C6x2. So G[{x1,x2,x3,x4,x5}{C1,C2,C4,C5,C6}] contains a subdivision of K3,3, a contradiction.

    Case (2.2). N(Ci){x1,x2,x3,x4} for i=1,2,3 but N(Cj){x1,x2,x3,x4} for 4j7.

    Then N(C3)={x1,x2,x4}. (The case that N(C3)={x1,x3,x4} is symmetric, and the proof is similar). As the neighbourhood of Ci (i=4,5,6,7) is not contained in {x1,x2,x3,x4}, Ci is adjacent to x5 for i=4,5,6,7. In G[{x1,x2,x3,x4}{C1,C2,C3}], any two of x1,x3,x4 are symmetric, and any two of C1,C2,C3 are symmetric. By the assumption of Case 2, we have the following four cases.

    Case (2.2.1). Suppose N(C4)={x3,x4,x5}, N(C5)={x1,x4,x5}, and N(C6)={x1,x3,x5}.

    Notice that any two of x1,x3,x4 are symmetric, since Case (2.1) does not happen, N(C7) is not contained in {x1,x3,x4,x5}, without loss of generality, assume that N(C7)={x2,x3,x5}. Now C2 is adjacent to x2,x3,x4, C1 is adjacent to x2,x3, and is connected to x4 by path C1x1C3x4, C7 is adjacent to x2,x3, and is connected to x4 by path C7x5C4x4. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C7}] contains a subdivision of K3,3, a contradiction.

    Case (2.2.2). Suppose N(C4)={x3,x4,x5}, N(C5)={x1,x3,x5}.

    Now x1 and x4 are symmetric. By the assumption of Case 2, as Case (2.2.1) does not hold, we have two cases: N(C6)={x2,x4,x5}; or N(C6)={x2,x3,x5}. Suppose N(C6)={x2,x4,x5}. Then C2 is adjacent to x2,x3,x4, C1 is adjacent to x2,x3, and is connected to x4 by path C1x1C3x4, C6 is adjacent to x2,x4, and is connected to x3 by path C6x5C5x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C5,C6}] contains a subdivision of K3,3, contradicting to the fact that G is planar. Suppose N(C6)={x2,x3,x5}. Then C2 is adjacent to x2,x3,x4, C1 is adjacent to x2,x3, and is connected to x4 by path C1x1C3x4, C6 is adjacent to x2,x3, and is connected to x4 by path C6x5C4x4. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction.

    Case (2.2.3). Suppose N(C4)={x3,x4,x5}.

    Now x3 and x4 are symmetric. We have two cases: (1) N(C5)={x2,x3,x5}, N(C6)={x2,x4,x5}; (2) N(C5)={x2,x3,x5}, N(C6)={x1,x2,x5}.

    In the first case, C2 is adjacent to x2,x3,x4, C1 is adjacent to x2,x3, and is connected to x4 by path C1x1C3x4, C6 is adjacent to x2,x4, and is connected to x3 by path C6x5C5x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C5,C6}] contains a subdivision of K3,3, a contradiction.

    In the second case, C1 is adjacent to x1,x2,x3, C2 is adjacent to x2,x3, and is connected to x1 by path C2x4C3x1, C6 is adjacent to x1,x2, and is connected to x3 by path C6x5C4x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction.

    There remains the last case in Case (2.2) as follows.

    Case (2.2.4). Suppose N(C4)={x2,x3,x5}, N(C5)={x2,x4,x5}, N(C6)={x1,x2,x5}.

    Now C1 is adjacent to x1,x2,x3, C2 is adjacent to x2,x3, and is connected to x1 by path C2x4C3x1, C6 is adjacent to x1,x2, and is connected to x3 by path C6x5C4x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C6}] contains a subdivision of K3,3, a contradiction.

    Case (2.3). Suppose only C1,C2 are adjacent only to vertices in {x1,x2,x3,x4}.

    Since N(Ci) is not contained in {x1,x2,x3,x4}, Ci is adjacent to x5 for i=3,4,,7. Now x1 and x4 are symmetric, x2 and x3 are symmetric in G[{x1,x2,x3,x4}{C1,C2}]. By the assumption of Case 2, each Ci is adjacent to exactly two vertices in {x1,x2,x3,x4} besides x5 (i=3,4,,7), and Ci and Cj (3i<j7) are not adjacent to the same two vertices in {x1,x2,x3,x4}. Since the number of combinations of two vertices in {x1,x2,x3,x4} is totally C(4,2)=4×32!=6, there is exactly one combination of two vertices which are not adjacent to the same Ci (3i7), and for each of the other combinations of two vertices in {x1,x2,x3,x4}, the two vertices are both adjacent to one Ci (3i7). Considering the symmetry of the roles of Ci (i=3,4,,7), there are six cases to take five combinations from the totally six combinations of two vertices in {x1,x2,x3,x4} such that the two vertices of each of the five combinations are both adjacent to a Ci (3i7). Also considering the symmetry of x1 and x4, and x2 and x3, there remains 3 cases as follows.

    Case (2.3.1). No Ci (3i7) is adjacent to both x1 and x4.

    Since the roles of C3,C4,,C7 are symmetric, without loss of generality, assume that N(C3)={x1,x2,x5}, N(C4)={x3,x4,x5}, N(C5)={x2,x4,x5}, N(C6)={x1,x3,x5}, N(C7)={x2,x3,x5}.

    Now C7 is adjacent to x2,x3,x5, C4 is adjacent to x3,x5, and is connected to x2 by path C4x4C2x2, C3 is adjacent to x2,x5, and is connected to x3 by path C3x1C6x3. So G[{x1,x2,x3,x4,x5}{C2,C3,C4,C6,C7}] contains a subdivision of K3,3, a contradiction.

    Case (2.3.2). No Ci (3i7) is adjacent to both x1 and x2. (For x1,x3; x2,x4; x3,x4, the discussion is similar.)

    Since the roles of C3,C4,,C7 are symmetric, without loss of generality, assume that N(C3)={x1,x4,x5}, N(C4)={x1,x3,x5}, N(C5)={x2,x4,x5}, N(C6)={x3,x4,x5}, N(C7)={x2,x3,x5}.

    Now C2 is adjacent to x2,x3,x4, C1 is adjacent to x2,x3, and is connected to x4 by path C1x1C3x4, C5 is adjacent to x2,x4, and is connected to x3 by path C5x5C7x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C5,C7}] contains a subdivision of K3,3, a contradiction.

    Case (2.3.3). No Ci (3i7) is adjacent to both x2 and x3.

    Since the roles of C3,C4,,C7 are symmetric, without loss of generality, assume that N(C3)={x1,x4,x5}, N(C4)={x1,x2,x5}, N(C5)={x1,x3,x5}, N(C6)={x2,x4,x5}, N(C7)={x3,x4,x5}.

    Now C1 is adjacent to x1,x2,x3, C2 is adjacent to x2,x3, and is connected to x1 by path C2x4C3x1, C4 is adjacent to x1,x2, and is connected to x3 by path C4x5C5x3. So G[{x1,x2,x3,x4,x5}{C1,C2,C3,C4,C5}] contains a subdivision of K3,3, a contradiction.

    Case 3. Suppose that, for any two vertices Ci and Cj (1i<j7), |N(Ci)N(Cj)|1 and Cases 1 and 2 do not hold.

    Since |N(Ci)|=|N(Cj)|=3 and |S|=5, |N(Ci)N(Cj)|=1 (1i<j7). Without loss of generality, assume that N(C1)={x1,x2,x3}, and N(C2)={x3,x4,x5}. Then N(C1)N(C2)={x3}. By the assumption of Case 3, |N(C3)N(C1)|=1, then there are two vertices of N(C3) which are not in {x1,x2,x3}, hence |N(C3)N(C2)|2, which contradicts the assumption of Case 3.

    In all cases discussed above, we can always obtain contradiction. So ω(GS)|S|+2 does not hold. By Lemma 1, when ν(G)2k, G is k-subconnected for k=4,5,6.

    Remark 1. Now we give some counterexamples to show the sharpness of Corollaries 7 and 8, and Theorem 10. Let H be a connected planar graph, let G1,G2,G3 be three copies of H, and let v be a vertex not in Gi (i=1,2,3). Let G be the graph such that v is joined to Gi (i=1,2,3) by an edge respectively. Then G is a 1-connected planar graph, but G is not 2-subconnected since we take a vertex vi in Gi (i=1,2,3) and let v4=v, then there are not two independent paths joining v1,v2,v3,v4 in two pairs in G. So Corollary 7 is sharp.

    Let H be a planar embedding of a 2-connected planar graph, let G1,G2,G3,G4 be four copies of H, let v5 and v6 be two vertices not in Gi (i=1,2,3,4). Let G be the graph such that v5 and v6 are joined to two different vertices on the outer face of Gi (i=1,2,3,4) by edges respectively. Then G is a 2-connected planar graph, but is not 3-subconnected since we take a vertex vi in Gi (i=1,2,3,4), then there are not three independent paths joining v1,v2,v3,v4,v5,v6 in three pairs. So Corollary 8 is sharp.

    Let G0 be a triangle with vertex set {v4,v5,v6}, then insert a vertex vi into a triangle inner face of Gi1 and join vi to every vertex on the face by an edge respectively to obtain Gi for i=1,2,3. Let G=G3. Notice that ν(G)=6, ε(G0)=3, and each time when we insert a vertex, the number of edges increases by 3. So ε(G)=3+3+3+3=12. By the Euler's Formula, ϕ=εν+2=126+2=8. Let H be a planar embedding of a 3-connected planar graph. Then put a copy of H into every face of G and join each vertex on the triangle face of G to a distinct vertex on the outer face of H to obtain a planar graph G. Then G is a 3-connected planar graph with ν(G)2k for k=7, but G is not 7-subconnected since we have a cutset S={v1,v2,v3,v4,v5,v6}, but GS has 8 copies of H (8 components) and ω(GS)=8|S|+2. By Lemma 1, the conclusion holds. So Theorem 10 is sharp.

    Remark 2. For a 3-connected planar graph G with at least 2k vertices, by Lemma 6, G is obviously k-subconnected for k=1,2,3.

    In the last section, we prove the k-subconnectivity of k-connected planar graphs for k=1,2,,5.

    Since a k-subconnected graph is a spanning substructure of a k-connected graph, in the future, we can work on the number of edges deleted from a k-connected graph such that the resulting graph is still k-subconnected.

    We may also extend the k-subconnectivity of planar graphs to find the subconnectivity of general graphs with a higher genus.



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