A structure theorem for strong immersions
A graph is strongly immersed in if is obtained from by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of are mapped to distinct vertices of (branch vertices) and edges of are mapped to pairwise edge-disjoint paths in , each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion.
In this paper, we consider graphs which can have parallel edges and loops, where each loop contributes to the degree of the incident vertex. A graph without parallel edges and loops is called simple.
Various containment relations between graphs have been studied in structural graph theory. The best known ones are perhaps minors and topological minors. A graph is a minor of if it can be obtained from by a sequence of edge and vertex removals and edge contractions. A graph is a topological minor of if a subdivision of is a subgraph of , or equivalently, if can be obtained from by a sequence of edge and vertex removals and by suppressions of vertices of degree two. In their fundamental series of papers, Robertson and Seymour developed the theory of graphs avoiding a fixed minor, giving a description of their structure  and proving that every proper minor-closed class of graphs is characterized by a finite set of forbidden minors . The topological minor relation is somewhat harder to deal with (and in particular, there exist proper topological minor-closed classes that are not characterized by a finite set of forbidden topological minors), but a description of their structure is also available [6, 4].
In this paper, we consider the related notion of a graph immersion. Let and be graphs. An immersion of in is a function from vertices and edges of such that
is a vertex of for each , and is injective.
is a connected subgraph of for each , and if is distinct from , then and are edge-disjoint.
If is incident with , then is a vertex of , and if is a loop, then contains a cycle passing through .
An immersion is strong if it additionally satisfies the following condition:
If is not incident with , then does not contain .
When we want to emphasize that an immersion does not have to be strong, we call it weak.
If is a topological minor of , then is also strongly immersed in . On the other hand, an appearance of as a minor does not imply an immersion of , and conversely, an appearance of as a strong immersion does not imply the appearance as a minor or a topological minor. Nevertheless, many of the results for minors and topological minors have analogues for immersions and strong immersions. For example, any simple graph with minimum degree at least contains a strong immersion of the complete graph (DeVos et al. ), as compared to similar sufficient minimum degree conditions for minors (, Kostochka , Thomason ) and topological minors (, Bollobás and Thomason , Komlós and Szemerédi ). Furthermore, every proper class of graphs closed on weak immersions is characterized by a finite set of forbidden immersions .
Let us restrict our attention for the moment to weak immersions. Fix to be a positive integer. It is easy to show that if a graph contains a set of vertices such that for every pair of vertices there does not exist an edge cut of order less than serparating and , then contains as a weak immersion. From this observation, we see that any graph which does not contain as a weak immersion must either have a small number of large degree vertices, or alternatively, there exists a small edge cut separating two big degree vertices. This gives rise to an easy structure theorem for weak immersions as shown in DeVos et al.  and Wollan .
The same is not true for strong immersions. There exist graphs which are arbitrarily highly edge connected and still have no strong immersion of (although such graphs will necessarily not be simple by the extremal result of  mentioned above). As an example, let be a positive integer and consider the graph obtained from from a path of length by adding parallel edges to each edge of the path. Then is edge connected but does not contain even as a strong immersion. The example can be made more complex as well. If we add edges to connect every pair of vertices at distance two on the path, the resulting graph will still have no strong immersion of . When the graph is assumed to be highly edge connected, this is essentially the only obstruction to a graph excluding a strong immersion of a fixed clique as shown by Marx and Wollan .
The main result of this article is a decomposition theorem for strong immersions. The basis is a theorem which says that if a graph avoids a strong immersion of a fixed clique and contains a large set of vertices which are pairwise highly edge connected, then the graph must have a decomposition with respect to yielding an obstruction to the existence of a strong clique immersion, similar in spirit to the construction of the highly edge connected graph avoiding a strong clique immersion described in the previous paragraph. From this decomposition theorem, it is straightforward to derive a structure theorem for graphs excluding a fixed graph as a strong immersion.
We first rigorously define the decomposition which arises. A near-partition of a set is a family of subsets , possibly empty, such that and for all . Let be a subset of vertices of a graph . A path-like decomposition of with respect to is an ordering , , …, of the vertices of and a near-partition , …, of . The elements of the near-partition are called bags of the decomposition. For a vertex , the -cut of the decomposition is the set of edges of with one endpoint in and the other endpoint in . The width of the decomposition is the maximum size of an -cut with . For a set , let denote the number of bags of the decomposition that intersect . For an integer , we say that is -bounded in the decomposition when . For an integer , we say that a set is -edge-connected if no two vertices of are separated by an edge-cut of size less than in . A set is -linear if there exists a set of size at most such that has a path-like decomposition with respect to of width less than and the neigborhood of every vertex of is -bounded in .
The following theorem is the main technical result of this article. It shows that every highly edge-connected set in a graph is -linear for some bounded value if does not contain a strong immersion of a fixed graph .
For every graph , there exists a value such that if a graph does not contain as a strong immersion, then every -edge-connected subset of is -linear.
Conversely, we show as well that the property of being -linear is a good approximate characterization of graphs excluding a fixed immersion in that every graph which satisfies this property for every highly edge connected set cannot contain a strong immersion of a big clique.
Assuming Theorem 1, we can derive a global structure theorem for graphs excluding a fixed graph as a strong immersion. The theorem and its proof are presented in Section 2. The proof of Theorem 1 is given in Section 3 and in the final section, we show that the converse statement to the decomposition and structure theorem are approximately true.
We establish some notation we will use going forward. Let be a graph and . We use to denote the subgraph of induced by . The graph refers to the subgraph of induced on . For a subset of edges, is the subgraph with vertex set and edge set . We will use to refer to the set of edges with one endpoint in and one endpoint not in (specifically, does not contain any loops). A separation of is a pair of non-empty subsets of such that and every edge has all it’s endpoints either contained in or contained in .
2 A structure theorem
In this section, we show how Theorem 1 gives rise to a global structure theorem for graphs which exclude a fixed graph as a strong immersion. We first present some further notation.
When studying graph minors, a natural decomposition is to break the graph on a small vertex cutset and look at the structure on each side of the cutset. This gives rise to the operation of clique sums on graphs. Given that graph immersions consist of a set of edge disjoint paths, it is natural to instead look at when the graph can be decomposed on a small edge cut. This motivates the definition of what we will refer to as edge sums in graphs.
Let , , and be graphs. Let be a positive integer. The graph is a -edge sum of and if the following holds. There exist vertices such that for and a bijection such that is obtained from by adding an edge from to for every pair of edges such that , , the ends of are and , and the ends of are and .
We will also refer to a -edge sum as an edge sum of order . The edge sum is grounded if there exist vertices and in and , respectively, such that for , and there exist edge-disjoint paths linking and . If can be obtained by a -edge sum of and , we write .
The following lemma appears in  and shows that edge sums preserve the presence of immersions.
Lemma 2 ().
Let , , and be graphs and let be a positive integer. Assume , and assume that the edge sum is grounded. Let be an arbitrary graph. If or admits an immersion of , then does as well. If the immersion in either or is strong, then the immersion in is also strong.
Just as clique sums give rise to tree decompositions, edge sums yield a natural tree-like decomposition of graphs.
A tree-cut decomposition of a graph is a pair such that is a tree and is a near-partition of the vertices of indexed by the vertices of . For each edge in , has exactly two components, namely and containing and respectively. The adhesion of the decomposition is
when has at least one edge, and 0 otherwise. The sets are called the bags of the decomposition.
Note that the definition allows bags to be empty.
We will need to define one more operation on graphs. Let be a graph and . The graph is obtained by consolidating if we identify the vertices of to a single vertex and delete all loops incident to .
Let be a graph and a tree-cut decomposition of . Fix a vertex . The torso of at is the graph defined as follows. If , then the torso of at is simply itself. If , let the components of be for some positive integer . Let for . Then is made by consolidating each set to a single vertex . The vertices are called the core vertices of the torso. The vertices are called the peripheral vertices of the torso. When there can be no confusion as to the graph in question, we will also refer to the torso of at a vertex .
The following lemma shows that tree cut decompositions can be combined in an edge sum of graphs.
Lemma 3 ().
Let , , and be graphs such that for some . If has a tree-cut decomposition for , then has a tree-cut decomposition such that the adhesion of is equal to
Moreover, for every , there exists and a vertex in such that the torso of at is isomorphic to the torso of at . Finally, every core vertex of is a core vertex of .
We can now state the structure theorem for graphs excluding a fixed clique immersion in terms of a tree-cut decomposition. The proof will follow easily assuming Theorem 1. We say that a graph is -basic if the set of all its vertices of degree at least is -linear, i.e., it has a path-like decomposition such that all the vertices in its bags have degree less than .
For every graph , there exists an integer such that if a graph does not contain as a strong immersion, then there exists a tree-cut decomposition of of adhesion less than such that each torso is -basic.
Proof (assuming Theorem 1).
Fix the graph and let be the value given in Theorem 1. Assume the statement is false, and let be a counterexample on a minimum number of edges. The set of vertices of degree at least in is not -edge-connected, as otherwise Theorem 1 yields a contradiction. Thus, we may assume that there exists vertices and each of degree at least such that there exists with , and .
Let be the graph obtained by consolidating and the graph obtained by consolidating . By construction, for some positive integer , and if we assume we chose a minimum order edge cut separating and , the edge sum is grounded. Given that both and have degree at least , we see that and . By Lemma 2, neither nor contains as a strong immersion. Both and have the desired decomposition by minimality, and therefore, by Lemma 3, has the desired decomposition as well. ∎
3 Proof of Theorem 1
We prove a slightly stronger statement which gives a clearer picture on the relationship between the parameters involved. Let be a graph and be positive integers. A set is -linear if there exists a set of size at most such that has a path-like decomposition with respect to of width less than and the neigborhood of every vertex of is -bounded in .
For every graph , there exist integers , and such that if a graph does not contain as a strong immersion, then every -edge-connected subset of is -linear.
To see that Theorem 5 is in fact a strengthening of Theorem 1, we observe the following. Assume that a subset of vertices in a graph is -linear for positive integers . Then a path-like decomposition of with respect to which certifies that is -linear trivially certifies as well that is -linear for all , , and . Thus, is -linear for , implying that Theorem 1 is an immediate consequence of Theorem 5.
For the remainder of this section, we define
We use the following result of Dvořák and Klimošová .
Theorem 6 ().
Let be a graph and . Let be an integer. Let be a set of vertices such that contains no edge cut of size less than separating from a vertex in . If a graph of maximum degree at most does not appear in a graph as a strong immersion, then there exist sets and such that and the component of that contains does not contain any vertex of .
For a graph , a set and an integer , let denote the graph with vertex set such that two vertices and in are adjacent in iff contains at least pairwise edge-disjoint paths joining with . As a corollary of Theorem 6, Dvořák and Klimošová  proved that if is sufficiently edge-connected and sufficiently large and avoids some fixed graph as a strong immersion, then is connected. We need a strenghtening of this claim.
Let be a graph and let . For all integers , there exists such that the following holds. Let be a graph, let be a -edge-connected set and let be a subset of of size at most . Suppose that is not connected and let be a separation of . If does not contain as a strong immersion, then contains an edge-cut of size less than separating from .
Let , and .
For , every vertex is separated from by an edge-cut of size less than in .
Suppose the claim is false. By symmetry, we can assume that . Apply Theorem 6 in for and , obtaining sets and , where , such that the component of that contains does not contain any vertex of . For each , apply Theorem 6 for and , obtaining sets and , where , such that the component of that contains does not contain any vertex of . Let and let , and note that and .
By Menger’s theorem, there exists a set of pairwise edge-disjoint paths from to in . Let consist of the paths that do not contain edges of ; we have . Consider a path . Let , , …, be the vertices of in order, where and . As the component of that contains does not contain any vertex of , the vertex belongs to . Let be the largest index such that belongs to . As the component of that contains does not contain any vertex of , it follows that belongs to . Consequently, contains a set of pairwise edge-disjoint paths joining vertices of with vertices of and otherwise disjoint from . By the pigeonhole principle, there exist vertices and contained in at least of these paths, and thus is an edge of . This contradicts the assumption that is a sepearation of . ∎
Consider any vertex . Since is -edge-connected, contains at least pairwise edge-disjoint paths from to . By Claim 1, at least of these paths pass through a vertex of . By pigeonhole principle, we have the following.
For every , there exists a vertex such that contains at least pairwise edge-disjoint paths from to .
Suppose that the lemma is false and that does not contain an edge-cut of size less than separating from . By Menger’s theorem, contains a set of pairwise edge-disjoint paths from to and otherwise disjoint from . By Claim 1, each vertex of is incident with less than of these paths, and thus we can select such that and the paths in have pairwise distinct ends in . Furthermore, we can select and of size at least such that every incident with a path in satisfies .
Let be the set of endpoints of the paths of in and note that . Consider any sets and such that . We have , and thus . Since , there exists a path ending in and disjoint with . Let be the endpoint of in . By Claim 2, there exists a path from to disjoint with and . Therefore, contains a path from to . Since this holds for every and with , we obtain a contradiction with Theorem 6. ∎
Furthermore, avoiding a strong immersion of a fixed graph restricts the structure of , as long as is large enough.
Let and be graphs, let be a subset of and let be an integer. If and does not contain as a strong immersion, then does not contain as a minor.
The claim is trivial if . Suppose that and that contains as a minor, that is, contains a subtree with leaves and at least one non-leaf vertex. Let be a non-leaf vertex of and let be the set of leaves of . Let be an injective function mapping to . By the definition of and Menger’s theorem, there exists a set of pairwise edge-disjoint paths in from to , such that every vertex is contained in exactly of these paths. A half-edge of is a pair , where is an edge of and is incident with . Note that there exists a bijective function from half-edges of to such that the path contains the vertex for every half-edge . We extend to a strong immersion of in by defining for every edge of . ∎
The next lemma gives an approximate characterization of when a graph does not contain a large minor. A set of vertices of a graph is a linearizing set if is a vertex-disjoint union of paths.
Lemma 9 ().
If a simple connected graph does not contain as a minor, then has a linearizing vertex set of size at most .
We now give the proof of Theorem 5.
Proof of Theorem 5.
Let , , and let be the corresponding constant from Lemma 7. Let and .
Let be a -edge-connected subset of , and let . If is not connected, then there exists a separation of . By Lemma 7 applied with , it follows that contains an edge-cut of size less than separating from . This is a contradiction, since is -edge-connected.
Therefore, is a connected simple graph, and by Lemma 8, does not contain as a minor. Let be the smallest linearizing set in . By Lemma 9, we have . Let , , …, be an ordering of such that for , the neighbors of in are contained in .
If , we set . If , we set and . If , we set and . In all the cases, we obtain a path-like decomposition of with respect to of width , and since , the neigborhood of every vertex of is -bounded. Therefore, assume that .
Consider an index such that and let and . A set is an -separator if , and . We set to be the number of edges of with one end in and the other end in . Let be an -separator with as small as possible, and subject to that with minimal. By Lemma 7 applied with , we have .
If , then .
Consider indices and such that . Let and . Note that is an -separator and is a -separator, and thus and . Observe that , where is the number of edges with one end in and the other end in , and is the number of edges incident with or and with the other end in . Consequently, . Putting the inequalities together, we have and . Since is chosen with minimal, it follows that . Therefore, . Furthermore, the inclusion is sharp, since . ∎
Let and . Let , and for , let us set . By Claim 3, , …, is a near-partition of , and thus the ordering , …, and the sets , …, form a path-like decomposition of with respect to . Since for , the width of is less than .
For , each component of contains a neighbor of or .
Suppose that is the vertex set of a component of containing neighbors of neither nor . We say that an edge of with one end in and the other end not in is backward if its end not in belongs to , and it is forward otherwise. Note than neither forward nor backward edges are incident with .
Let and note that is an -separator. Since , the choice of implies that , and thus there are more backward edges than forward ones. Since and induces a component of containing no neighbors of , all backward edges are incident with vertices in . Since , we have . However, then is an -separator with , which is a contradiction. ∎
To complete the proof, we must show that every vertex of is -bounded in . Suppose that the neighborhood of a vertex is not -bounded. By Claim 4, contains at least paths from to vertices of whose vertex sets pairwise intersect only in . Let denote the set of their endpoints in . Suppose that sets and have the property that the component of that contains does not contain any vertex of . Then each of the paths from to contains either a vertex of or an edge of , and thus . This contradicts Theorem 6. ∎
4 An approximate converse
We conclude by showing that the decomposition guaranteed by Theorem 5 does indeed give a good approximation of graphs excluding strong clique immersions.
For all integers , , and , there exists an integer such that if every -edge-connected subset of is -linear, then does not contain as a strong immersion.
Let , and suppose that is a strong immersion of in . Let be the vertex set of and let . Since , is -edge-connected in , and thus it is -linear. Let be a subset of of size at most such that has a path-like decomposition with respect to of width less than and the neigborhood of every vertex of is -bounded in . Let , …, be the ordering of according to and let , …, be the bags of . Since and the neighborhood of every vertex of in is -bounded in , there exists an index such that the vertices of have no neighbors in . Let be the union of the -cut and the -cut of . Consider an edge of incident with such that is incident neither with nor . Since the immersion is strong, contains neither nor , and thus contains an edge of . Therefore, there are at most such edges incident with , which is a contradiction since . ∎
Similarly, the structure described in Theorem 4 is sufficient to exclude a large clique as a strong immersion.
For every integer there exists an integer such that if a graph has a tree-cut decomposition of adhesion less than such that each torso is -basic, then does not contain as a strong immersion.
Let , and suppose that is a strong immersion of in . Let be the vertex set of and let . The set is -edge-connected in , and since has adhesion less than , we conclude that there exists such that . Let be the torso of at , and observe that also contains as a strong immersion. However, we can now obtain a contradiction in the same way as in the proof of Theorem 10. ∎
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