Forbidding Kuratowski Graphs as Immersions
The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph contains a graph as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely and , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive -edge-sums, for , starting from graphs that are planar sub-cubic or of branch-width at most 10.
Keywords: graph immersions, Kuratowski graphs, tree-width, branch-width.
A famous graph theoretic result is the theorem of Kuratowski which states that a graph is planar if and only if it does not contain and (also known as the Kuratowski graphs) as a topological minor, that is, if and cannot be obtained from the graph by applying vertex and edge removals and edge dissolutions. It is well-known that the topological minor relation defines a (partial) ordering of the class of graphs.
In a similar way, the immersion and the minor orderings can be defined in graphs if instead of vertex dissolutions we ask for edge lifts and edge contractions respectively. (For detailed definitions see Section 2.) Notice here that the topological minor ordering is stronger than the minor and the immersion orderings in the sense that if a graph contains a graph as a topological minor then it also contains it as an immersion and as a minor but the inverse direction does not always hold.
In the celebrated Graph Minors theory, developed by Robertson and Seymour, it was proven that both the immersion and minor orderings are well-quasi-ordered, that is, there are no infinite sets of mutually non-comparable graphs [16, 19] according to these orderings. This result has as a consequence the complete characterization of the graph classes that are closed under taking immersions or minors in terms of forbidden graphs, where a graph class is closed under taking immersions (respectively minors) if for any graph that belongs to the graph class all of its immersions (respectively minors) also belong to the graph class. For example, by an extension of the Kuratowski theorem (also known as Wagner’s theorem), it is also known that a graph is planar if and only if it does not contain and as a minor.
Thus, a question that naturally arises is about the characterization of the structure of a graph that excludes some fixed graph as an immersion or as a minor. While this subject has been extensively studied for the minor ordering (see [6, 20, 18, 15, 9, 2, 3, 13, 17, 22]), the immersion ordering only recently attracted the attention of the research community [10, 4, 8, 1, 12]. In , DeVos et al. proved that for every positive integer , every simple graph of minimum degree at least contains the complete graph on vertices as a (strong) immersion and in  Ferrara et al., given a graph , provide a lower bound on the minimum degree of any graph in order to ensure that is contained in as an immersion. More recently, in , Seymour and Wollan proved a structure theorem for graphs excluding complete graphs as immersions.
In terms of graph colorings, Abu-Khzam and Langston in  provide evidence supporting the analog of Hadwiger’s Conjecture according to the immersion ordering, that is, the conjecture stating that if the chromatic number of a graph is at least then contains the complete graph on vertices as an immersion and prove it for . This conjecture is proven for and by Lescure and Meyniel in  and by DeVos et al. in  independently. The most recent result on colorings is an approximation of the list coloring number on graphs excluding the complete graph as immersion .
Finally, in terms of algorithms, in , Grohe et al. gave a cubic time algorithm that decides whether a fixed graph is contained in an input graph as immersion and in  it was proved that the minimal graphs not belonging to a graph class closed under immersions can be computed when an upper bound on their tree-width and a description of the graph class in Monadic Second Order Logic are given.
In this note we characterize the structure of the graphs that do not contain and as immersions. As these graphs already exclude Kuratowski graphs as topological minors they are already planar. Additionally, we show that they have a more special structure: they can be constructed by repetitively, joining together simpler graphs, starting from either graphs of small decomposability or by planar graphs with maximum degree 3. In particular, we prove that a graph that does not contain neither nor as immersions can be constructed by applying consecutive -edge-sums, for , to graphs that are planar sub-cubic or of branch-width at most 10.
For every integer , we let . All graphs we consider are finite, undirected, and loopless but may have multiple edges. Given a graph we denote by and its vertex and edge set respectively. Given a set (resp. ), we denote by (resp. ) the graph obtained from if we remove the edges in (resp. the vertices in along with their incident edges). We denote by the set of the connected components of . Given a vertex , we also use the notation . The neighborhood of a vertex , denoted by , is the set of edges in that are adjacent to . We denote by the set of the edges of that are incident to . The degree of a vertex , denoted by , is the number of edges that are incident to it, i.e., . Notice that, as we are dealing with multigraphs, . The minimum degree of a graph , denoted by , is the minimum of the degrees of the vertices of , that is, . A graph is called sub-cubic if all its vertices have degree at most 3. We also denote by the complete graph on vertices and by the complete bipartite graph with vertices in its one part and in the other. Let be a path and . We denote by the sub-path of with end-vertices and . Given two paths and who share a common endpoint , we say that they are well-arranged if their common vertices appear in the same order in both paths.
We say that a graph is a subgraph of a graph if can be obtained from , after removing edges or vertices. An edge cut in a graph is a non-empty set of edges that belong to the same connected component of and such that has more connected components than . If has one more connected component than then we say that is a minimal edge cut. Let be an edge cut of a graph and let be the connected component of containing the edges of . We say that is an internal edge cut if it is minimal and both connected components of contain at least 2 vertices. An edge cut is also called -edge-cut if it has cardinality .
In this paper we mostly deal with lanai graphs, that is graphs that are embedded in the sphere . We call such a graph, along with its embedding, -embeddable graph. Let , be two disjoint cycles in a -embeddable graph . Let also be the open disk of that does not contain points of , . The annulus between and is the set and we denote it by . Notice that is a closed set. If is a collection of cycles of a -embeddible graph . We say that is nested if for every , .
Contractions and minors.
The contraction of an edge from is the removal from of all edges incident to or and the insertion of a new vertex that is made adjacent to all the vertices of such that edges corresponding to the vertices in increase their multiplicity, that is, if there was a vertex , edges joining and and, edges joining and then in the resulting graph there will be edges joining with . Finally, remove any loops resulting from this operation. Given two graphs and , we say that is a contraction of , denoted by , if can be obtained from after a (possibly empty) series of edge contractions. Moreover, is a minor of if is a contraction of some subgraph of .
A subdivision of a graph is any graph obtained after replacing some of its edges by paths between the same endpoints. A graph is a topological minor of (denoted by ) if contains as a subgraph some subdivision of .
The lift of two edges and to an edge is the operation of removing and from and then adding the edge in the resulting graph. We say that a graph can be (weakly) immersed in a graph (or is an immersion of ), denoted by , if can be obtained from a subgraph of after a (possibly empty) sequence of edge lifts. Equivalently, we say that is an immersion of if there is an injective mapping such that, for every edge of , there is a path from to in and for any two distinct edges of the corresponding paths in are edge-disjoint, that is, they do not share common edges. Additionally, if these paths are internally disjoint from , then we say that is strongly immersed in (or is a strong immersion of ). The injective mapping together with the edge-disjoint paths is called a model of in defined by .
Let and be graphs, let be vertices of and respectively, and consider a bijection where . We define the -edge sum of and on and as the graph obtained if we take the disjoint union of and , identify with , and then, for each , lift and to a new edge and remove the vertex . (See Figures 1 and 2)
Let be a graph, let be a minimal -edge cut in , and let be the connected component of that contains . Let also and be the two connected components of . We denote by the graph obtained from after contracting all edges of to a single vertex . We say that the graph consisting of the disjoint union of the graphs in is the -split of and we denote it by . Notice that if is connected and is a minimal -edge cut in , then is the result of the -edge sum of the two connected components and of on the vertices and . From Menger’s Theorem we obtain the following.
Let be a positive integer. If is a connected graph that does not contain an internal -edge cut, for some and are distinct vertices such that then there exist edge-disjoint paths from to .
If is a -immersion free connected graph and is a minimal internal -edge cut in , for then both connected components of are -immersion free.
For contradiction assume that is a -immersion free connected graph and one of the connected components of , say , contains or as an immersion, where is a minimal internal -edge cut in , . Assume that is immersed in and let be a model of in . Let also be the newely introduced vertex of . Notice that if and is not an internal vertex of any of the edge-disjoint paths between the vertices in , then is a model of in . As , is a model of in , a contradiction to the hypothesis. Thus, we may assume that either or is an internal vertex in at least one of the edge-disjoint paths between the vertives in . Note that, as neither nor contain vertices of degree , or .
We first exclude the case where , that is, only appears as an internal vertex on the edge-disjoint paths. Observe that, as , belongs to exactly one path in the model defined by . Let and be the neighbors of in . Recall that, by the definition of an internal -split, there are vertices and in such that . Furthermore, as is connected, there exists a -path in . Therefore, be substituting the subpath by the path defined by the union of the edges and the path in we obtain a model of in defined by , a contradiction to the hypothesis.
Thus, the only possible case is that . As and , defines a model of in . Let and be the neighbors of in . We claim that there is a vertex in and edge-disjoint paths from to in , thus proving that there exists a model of in as well, a contradiction to the hypothesis. By the definition of an internal -split, there are vertices and in such that , . Recall that is connected. Therefore, if for every vertex , , contains a path whose endpoints, say and belong to and internally contains the vertex in , say . Then it is easy to verify that satisfies the conditions of the claim. Assume then that there is a vertex of degree at least 3. Let be the graph obtained from after removing all vertices in and adding a new vertex that we make it adjacent to the vertices in . As does not contain an internal -edge cut, , does not contain an internal -edge cut, . Therefore, from Observation 2.1 and the fact that , we obtain that there exist 3 edge-disjoint paths from to in and thus in . This completes the proof of the claim and the lemma follows. ∎
Let and . A -cylinder, denoted by , is the cartesian product of a cycle on vertices and a path on vertices. (See, for example, Figure 3) A -railed annulus in a graph is a pair such that is a collection of nested cycles that are all met by a collection of paths (called rails) in a way that the intersection of a rail and a path is always a (possibly trivial, that is, consisting of only one vertex) path. (See, for example, Figure 3) Notice that given a graph embedded in the sphere and a -cylinder (-railed annulus respectively) of , then any two cycles of the -cylinder (-railed annulus respectively) define an annulus between them.
A branch decomposition of a graph is a pair , where is a ternary tree and is a bijection of the edges of to the leaves of , denoted by . Given a branch decomposition , we define as follows.
Given an edge let and be the trees in Then . The width of a branch decomposition is and the branch-width of a graph , denoted by , is the minimum width over all branch decompositions of . In the case where the width of the branch decomposition is defined to be 0. The following has been proven in .
If is a planar graph and are integers with and then either contains the -cylinder as a minor or has branch-width at most .
We now prove the following.
If is a planar graph of branch-width at least 11, then contains a (4,4)-railed annulus.
Let be a planar graph of branch-width at least 11. Then by Theorem 2.3, contains -cylinder as a minor. By the definition of the minor relation, contains a -railed annulus. ∎
Let be a graph embedded in some surface and let . We define a disk around as any open disk with the property that each point in is either or belongs to the edges incident to . Let and be two edge-disjoint paths in . We say that and are confluent if for every , that is not an endpoint of or , and for every disk around , one of the connected components of the set does not contain any point of . We also say that a collection of paths is confluent if the paths in it are pairwise confluent.
Moreover, given two edge-disjoint paths and in we say that a vertex that is not an endpoint of or is an overlapping vertex of and if there exists a around such that both connected components of contain points of . For a family of paths , a vertex of a path is called an overlapping vertex of if there exists a path such that is an overlapping vertex of and .
3 Preliminary results on the confluency of paths
Let be a graph and such that there exist edge-disjoint paths and from to and respectively. If the paths and are not well-arranged then there exist edge-disjoint paths and from to and respectively such that .
Let , where is the order that the vertices in appear in and, is the order that they appear in . As the paths are not well-arranged there exists such that . Without loss of generality assume that is the smallest such integer. Without loss of generality assume also that . We define
and observe that and satisfy the desired properties. (For an example, see Figure 4).
Before proceeding to the statement and proof of the next proposition we need the following definition. Given a collection of paths in a graph , we define the function such that is the number of pairs of paths for which is an overlapping vertex. Let
Notice that for every and thus . Observe also that if and only if is a confluent collection of paths.
Lemma 3.1 allows us to prove the main result of this section. We state the result for general surfaces as the proof for this more general setting does not have any essential difference than the case where is the sphere .
Let be a positive integer. If is a graph embedded in a surface , and is a collection of edge-disjoint paths from to in , then contains a confluent collection of well-arranged edge-disjoint paths from to where and such that .
Let be the spanning subgraph of induced by the edges of the paths in and let be a minimal spanning subgraph of that contains a collection of edge-disjoint paths from to . Let also be the collection of edge-disjoint paths from to in for which is minimum. It is enough to prove that .
For a contradiction, we assume that and we prove that there exists a collection of edge-disjoint paths from to in such that . As , then there exists a path, say , that contains an overlapping vertex . Let be the endpoint of which is different from . Without loss of generality we may assume that is the overlapping vertex of that is closer to in . Then there is a -path such that is an overlapping vertex of and . Let , and for every . As Lemma 3.1 and the edge-minimality of imply that the paths and are well-arranged, we obtain that is a path from to , . Let be . It is easy to verify that is a collection of edge-disjoint paths from to . We will now prove that .
First notice that if , then . Thus, it is enough to prove that .
Observe that if and is an overlapping vertex of and then is also an overlapping vertex of and . Furthermore, while is an overlapping vertex in the case where , it is not an overlapping vertex of and . It remains to examine the case where . In other words, we examine the case where one of the paths and , say , is or , and . Let be a disk around and be the two distinct disks contained in the interior of after removing .
We distinguish the following cases.
Case 1. is neither an overlapping vertex of and , nor of and (see Figure 5). Then it is easy to see that the same holds for the pairs of paths and and, and . Indeed, notice that for every , intersects exactly one of and . Furthermore, as is an overlapping vertex of and , both paths intersect the same disk. From the observation that , we obtain that is neither an overlapping vertex of and nor of and .
Case 2. is an overlapping vertex of and but not of and , (see Figure 6). Notice that exactly one of the following holds.
intersects exactly one of the disks or , say . Then intersects and intersects . Therefore, it is easy to see that, is not an overlapping vertex of and anymore but becomes an overlapping vertex of and .
intersects exactly one of the disks or , say . Then intersects and intersects . Therefore, it is easy to see that remains an overlapping vertex of and and does not become an overlapping vertex of and .
Case 3. is an overlapping vertex of both and and, and (see Figure 7). As above, exactly one of the following holds.
intersects exactly one of the disks or , say . Then intersects . It follows that is an overlapping vertex of both and and, and
intersects exactly one of the disks or , say . Then intersects . It follows that is neither an overlapping vertex of and nor of and .
From the above cases we obtain that and therefore , contradicting the choice of . This completes the proof of the Proposition. ∎
4 A decomposition theorem
We prove the following decomposition theorem for -immersion free graphs.
If is a graph not containing or as an immersion, then can be constructed by applying consecutive -edge sums, for , to graphs that either are sub-cubic or have branch-width at most 10.
Observe first that a -immersion-free graph is also -topological-minor-free, therefore, from Kuratowski’s theorem, is planar. Applying Lemma 2.2, we may assume that is a -immersion-free graph without any internal -edge cut, . It is now enough to prove that is either planar sub-cubic or has branch-width at most 10. For a contradiction, we assume that and that contains some vertex of degree . Our aim is to prove that contains as an immersion. First, let be the graph obtained from after subdividing all of its edges once. Notice that contains as an immersion if and only if contains as an immersion. Hence, from now on, we want to find in as an immersion.
From Lemma 2.4 , and thus , contains a -railed annulus as a subgraph. Observe then that also contains as a subgraph a -railed annulus such that the vertex of degree does not belong in the annulus between its cycles (Figure 8 depicts the case where is inside the annulus between the second and the third cycle). We denote by and the nested cycles and by and the rails of the above -railed annulus. Let be the annulus between and . Without loss of generality we may assume that separates from and that is edge-minimal, that is, there is no other annulus such that and .
Let now be the connected components of .
For every and every , .
Proof of Claim 1.
Indeed, assume the contrary. Then there is a cycle such that and define an annulus with and , a contradiction to the edge-minimality of the annulus . ∎
For every , we denote by and the unique neighbor of in and respectively (whenever they exist). We call the connected components that have both a neighbor in and a neighbor in substantial. Let . That is, is the set of graphs induced by the substantial connected components and their neighbors in the cycles and . Note that every edge of has been subdivided in and thus every edge for which and corresponds to a substantial connected component in .
We now claim that there exist four confluent edge-disjoint paths and from to in . This follows from the facts that does not contain an internal -edge cut, contains at least vertices, combined with Observation 2.1. Moreover, from Proposition 3.2, we may assume that and are confluent.
Let be the subpath of , where is the vertex in whose distance from in is minimum, . Recall that all edges of have been subdivided in . This implies that there exist four (possibly not disjoint) graphs in , say and such that , .
We distinguish two cases.
Case 1. The graphs and are vertex-disjoint.
This implies that the endpoints of and are disjoint. Let be the graph induced by the cycles and the paths and let and be confluent edge-disjoint paths from to , , and in such that
is minimum, that is, the number of the edges of the paths that is outside of is minimum, and
subject to i, is minimum.
Let also be the graph induced by , , , , and . From now on we work towards showing that contains as an immersion. For every we call a connected component of non-trivial if it contains at least an edge.
For every , contains at most one non-trivial connected component and is an endpoint of .
Proof of Claim 2.
First, notice that any path from to in contains and thus, . Observe now that is a subpath of whose internal vertices do not belong to , thus if belongs to a non-trivial connected component of , then is an endpoint of . We will now prove that any non-trivial connected component of contains . Assume in contrary that there exists a non-trivial connected component of that does not contain . Let be the endpoint of for which is minimum. Let also be the vertex in such that is minimum. Let be the subpath of with endpoints such that is a cycle with . We further assume that the interior of is the open disk that does not contain any vertices of . We will prove that for every path , . As this trivially holds for we will assume that . Observe that, for every , as for every connected component of it holds that . Furthermore, observe that is a separator in . This implies that does not belong to the interior of . Thus, if there is a vertex such that , there is a vertex , a contradiction to the confluency of the paths. We may then replace by , a contradiction to i. ∎
We denote by the endpoint of that is different from if