Excluding Graphs as Immersions in Surface Embedded Graphs
We prove a structural characterization of graphs that forbid a fixed graph as an immersion and can be embedded in a surface of Eüler genus . In particular, we prove that a graph that excludes some connected graph as an immersion and is embedded in a surface of Eüler genus has either “small” treewidth (bounded by a function of and ) or “small” edge connectivity (bounded by the maximum degree of ). Using the same techniques we also prove an excluded grid theorem on bounded genus graphs for the immersion relation.
Keywords: Surface Embeddable Graphs, Immersion Relation, Treewidth, Edge Connectivity.
A graph is an immersion of a graph if it can be obtained from by removing vertices or edges, and splitting off adjacent pairs of edges. The class of all graphs was proved to be well-quasi-ordered under the the immersion relation by Robertson and Seymour in the last paper of their Graph Minors series . Certainly, this work was mostly dedicated to minors and not immersions and has been the source of many theorems regarding the structure of graphs excluding some graph as a minor. Moreover, the minor relation has been extensively studied the past two decades and many structural results have been proven for minors with interesting algorithmic consequences (see, for example, [19, 18, 17, 4, 14, 21]). However, structural results for immersions started appearing only recently. In 2011, DeVos et al. proved that if the minimum degree of a graph is then contains the complete graph on vertices as an immersion . In  Ferrara et al., provided a lower bound (depending on graph ) on the minimum degree of a graph that ensures that is contained in as an immersion. Furthermore, Wollan recently proved a structural theorem for graphs excluding complete graphs as immersions as well as a sufficient condition such that any graph which satisfies the condition admits a wall as an immersion . The result in  can be seen as an immersion counterpart of the grid exclusion theorem , stated for walls instead of grids and using an alternative graph parameter instead of treewidth.
In terms of graph colorings, Abu-Khzam and Langston in  provided evidence supporting the immersion ordering analog of Hadwiger’s Conjecture, 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 proved it for . For , see [15, 6]. For algorithmic results on immersions, see [13, 2, 10, 12].
In this paper, we prove structural results for the immersion relation on graphs embeddable on a fixed surface. In particular, we show that if is a graph that is embeddable on a surface of Eüler genus and is a connected graph then one of the following is true: either has bounded treewidth (by a function that depends only on and ), or its edge connectivity is bounded by the maximum degree of , or it contains as a (strong) immersion. Furthermore, we refine our results to obtain a counterpart of the grid exclusion theorem for immersions. In particular, we prove (Theorem 3) that there exists a function such that if is a -edge-connected graph embedded on a surface of Eüler genus and the treewidth of is at least , then contains the -grid as an immersion. Notice that the edge connectivity requirement is necessary here as big treewidth alone is not enough to ensure the existence of a graph with a vertex of degree 4 as an immersion. Although a wall of height at least has treewidth at least , it does not contain the complete graph on vertices as an immersion, for any . Finally, our results imply that when restricted to graphs of sufficiently big treewidth embeddable on a fixed surface, large edge connectivity forces the existence of a large clique as an immersion.
Our result reveals several aspects of the behavior of the immersion relation on surface embeddable graphs. The proofs exploit variants of the grid exclusion theorem for surfaces proved in  and  and the results of Biedl and Kaufmann  on optimal orthogonal drawings of graphs.
For every positive integer , let denote the set . A graph is a pair where is a finite set, called the vertex set and denoted by , and is a set of 2-subsets of , called the edge set and denoted by . If we allow to be a multiset then is called a multigraph. Let be a graph. For a vertex , we denote by its (open) neighborhood, that is, the set of vertices which are adjacent to , and by the set of edges containing . Notice that if is a multigraph . The degree of a vertex is . We denote by the maximum degree over all vertices of .
If (respectively or or ) then (respectively or or ) is the graph obtained from by the removal of vertices of (respectively of vertex or edges of or of the edge ). We say that a graph is a subgraph of a graph , denoted by , if can be obtained from after deleting edges and vertices.
We say that a graph is an immersion of a graph (or is immersed in ), , 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. The function is called a model of in .
Let be a path and . We denote by the subpath of with endvertices 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.
A tree decomposition of a graph is a pair , where is a tree and is a function that maps every vertex to a subset of such that:
for every edge of there exists a vertex in such that , and
for every , if and , then for every vertex on the unique path between and in , .
The width of a tree decomposition is width and the treewidth of a graph is the minimum over the width, where is a tree decomposition of .
A surface is a compact 2-manifold without boundary (we always consider connected surfaces). Whenever we refer to a -embedded graph we consider a 2-cell embedding of in . To simplify notations, we do not distinguish between a vertex of and the point of used in the drawing to represent the vertex or between an edge and the line representing it. We also consider a graph embedded in as the union of the points corresponding to its vertices and edges. That way, a subgraph of can be seen as a graph , where in . Recall that is an open (respectively closed) disc if it is homeomorphic to (respectively ). The Eüler genus of a non-orientable surface is equal to the non-orientable genus (or the crosscap number). The Eüler genus of an orientable surface is , where is the orientable genus of . We refer to the book of Mohar and Thomassen  for more details on graphs embeddings. The Eüler genus of a graph (denoted by ) is the minimum integer such that can be embedded on a surface of the Eüler genus .
Let and be positive integers where . The -grid is the Cartesian product of two paths of lengths and respectively. A wall of height , , is the graph obtained from a -grid with vertices , , , after the removal of the “vertical” edges for odd , and then the removal of all vertices of degree 1. We denote such a wall by . The corners of the wall are the vertices , , and . (The square vertices in Figure 1.)
A subdivided wall of height is a wall obtained from after replacing some of its edges by paths without common internal vertices. We call the resulting graph a subdivision of and the new vertices subdivision vertices. The non-subdivision vertices are called original. The perimeter of a subdivided wall (grid) is the cycle defined by its boundary.
Let be a subdivided wall in a graph and be the connected component of that contains . The compass of in is the graph . Observe that is a subgraph of and is connected.
The layers of a subdivided wall of height are recursively defined as follows. The first layer of , denoted by , is its perimeter. For , the -th layer of , denoted by , is the -th layer of the subwall obtained from after removing from its perimeter and (recursively) all occurring vertices of degree 1 (see Figure 1).
We denote by the annulus defined by the cycles and , that is, by -th and -th layer, . Given an annulus defined by two cycles and , we denote by the interior of , that is, .
A subdivided wall of height is called tight if
the closed disk defined by the innermost (-th) layer of is edge-maximal (for reasons of uniformity we will denote this disk by ),
for every the annulus is edge-maximal under the condition that is edge-maximal.
If is a subdivided wall of height , we call brick of any facial cycle whose non-subdivided counterpart in has length 6. We say that two bricks are neighbors if their intersection contains an edge.
Let be a wall. We denote by the shortest path connecting vertices and and call these paths the horizontal paths of . Note that these paths are vertex-disjoint. We call the paths and the southern path of and northern part of respectively.
Similarly, we denote by the shortest path connecting vertices and with the assumption that for, , contains only vertices with . Notice that there exists a unique subfamily of of vertical paths with one endpoint in the southern path of and one in the northern path of . We call these paths vertical paths of and denote them by , , where and . (See Figure 2.)
The paths and are called the western part of and the eastern part of respectively. Note that the perimeter of the wall can alternatively be defined as the cycle .
Notice now that each vertex , is contained in exactly one vertical path, denoted by , and in exactly one horizontal path, denoted by , of . If is a subdivision of , we will use the same notation for
the paths obtained by the subdivisions of the corresponding paths of , with further assumption that is an original vertex of .
Given a wall and a layer of , different from the perimeter of . Let be the subwall of with perimeter . is also called the subwall of defined by . We call the following vertices, important vertices of ; the original vertices of that belong to and have degree 2 in the underlying non-subdivided wall of but are not the corners of (where we assume that shares the original vertices of ). (See Figure 3)
A layer of a wall that is different from its perimeter and defines a subwall of of height contains exactly important vertices.
Let be a graph embedded in a surface of Eüler genus . If , contains as a subgraph a subdivided wall of height , whose compass in is embedded in a closed disk .
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 . (See, Figure 4.) 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 .
An orthogonal drawing of a graph in a grid is a mapping which maps vertices to subgrids (called boxes) such that for every with , , and edges to -paths whose internal vertices belong to , their endpoints (called joining vertices of ) belong to the perimeter of , , and for every two disjoint edges , , the corresponding paths are edge-disjoint.
We need the following result.
Lemma 2 ().
If is a simple graph then it admits an orthogonal drawing in an -grid. Furthermore, the box size of each vertex is .
3 Preliminary Combinatorial Lemmata
Before proving the main result of this section we first state the following lemma which we will need later on.
Lemma 3 ().
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 edge-dsjoint paths from to such that .
Detachment tree of in .
Let be a graph embedded in a closed disk , be distinct vertices of , and be a family of confluent edge-disjoint paths such that is a path from to , . Let also be an internal vertex of at least two paths in . Let , , , denote the family of paths in that contain and be a disk around . Given any edge with we denote by its common point with the boundary of . Moreover, we denote by and the edges of incident to , .
We construct a tree in the following way and call it detachment tree of in . Consider the outerplanar graph obtained from the boundary of by adding the edges , . We subdivide the edges , , resulting to a planar graph. For every bounded face of the graph, let denote the set of vertices that belong to . We add a vertex in its interior and we make it adjacent to the vertices of . Finally we remove the edges that lie in the boundary of . We call this tree . Notice that for every with , the vertex is a leaf of . (See Figure 5.)
We replace by in the following way. We first subdivide every edge incident to , and denote by the vertex added after the subdivision of the edge . We denote by the resulting graph. Consider now the graph (where, without loss of generality, we assume that ). The graph is called the graph obtained from by replacing with .
Let be positive integers and be a multigraph containing as a subgraph a subdivided wall of height , whose compass is embedded in a closed disk . Furthermore, let , , , be vertices of such that there exists a confluent family of edge-disjoint paths from to the vertices , . Finally, let belonging to more than one paths of . The graph obtained from by replacing with contains as a subgraph a subdivided wall of height , whose compass is embedded in and there exists a family of confluent edge-disjoint paths from to , , in whose paths avoid .
Notice first that it is enough to prove the observation for the case where . Let , (and possibly ) be the edges incident to that also belong to . Notice now that the vertices , (and ) are leaves of . Thus, from a folklore result, there exists a vertex such that there exist 2 (or 3) internally vertex-disjoint paths from to and (and possibly ). ∎
We now state the following auxiliary definitions. Let be a multigraph that contains a wall of height whose compass is embedded in a closed disk. Let , that is, let be a vertex contained in the closed disk defined by the innermost layer of , and let be a path from to the perimeter of . For each layer of the wall, , we denote by the first vertex of (starting from ) that also belongs to and we call it incoming vertex of in .
We denote by the maximal subpath of that contains and is entirely contained in the wall defined by . Moreover, we denote by its endpoint in and call it outgoing vertex of in . Notice that and are not necessarily distinct vertices.
Let and be positive integers. Let be a graph and be a tight subdivided wall of of height , whose compass is embedded in a closed disk . Let also be a vertex such that . If there exist vertex-disjoint paths , , from to vertices of the perimeter then there is a brick of with that contains both and .
Assume the contrary. Then it is easy to see that we can construct an annulus such that and , a contradiction to the tightness of the wall. (See Figure 6.) ∎
Let be a positive integer and be a multigraph that contains as a subgraph a subdivided wall of height at least , whose compass is embedded in a closed disk . Let also be a set of vertices lying in the perimeter of , whose mutual distance in the underlying non-subdivided wall is at least 2. If there exist a vertex and internally vertex-disjoint paths from to vertices of , then there exist internally vertex-disjoint paths from to the vertices of in .
Assume first, without loss of generality, that the wall is tight. Let then be the paths from to and let be the clockwise cyclic ordering according to which they appear in . Our objective is to reroute the paths , so that they end up to the vertices of . To do so our first step is to identify a layer of the wall for which there exist two consecutive paths whose incoming vertices on the layer are “sufficiently far apart”.
Let . Consider the layer and for every let denote the path of starting from and ending in (considered clockwise), that is, the path of starting from the incoming vertex of in and ending to the incoming vertex of in , where in the case we abuse notation and assume that (see Figure 7). Let also be the index such that the path contains the maximum number of important vertices amongst the ’s. Without loss of generality we may assume that . From Observation 1, as defines a subwall of of height , contains exactly important vertices. Thus, at least important vertices are internally contained in . This concludes the first step of the proof.
Let now . At the next step, using the part of the wall that is contained in , that is, in the annulus between the -th and the -th layer of the wall, we find internally vertex-disjoint paths from the incoming vertices of the paths in to consecutive important vertices of the -th layer of the wall. These are the paths that will allow us to reroute the original paths.
Continuing the proof, let be a set of successive important vertices appearing clockwise in such that the paths and internally contain at least important vertices. Notice that, without loss of generality, we may assume that the vertices , , belong to the northern part of . Recall here that each original vertex of is contained in exactly one vertical path of . For every we assign the path to the vertex in the following way. Let be the maximal subpath of whose endpoints are and the important vertex of that also belongs to , which from now on we will denote by . Note here that the paths , , are vertex-disjoint and do not contain any of the vertices belonging to the interior of the disk defined by in the compass of (See, for example, Figure 8).
Notice now that , and thus , contains a path from to and a path from to that are vertex-disjoint and do not contain vertices of any path other than and . Consider now the consecutive layers of preceeding , that is, the layers , . For every let be the first time the path meets starting from and be the first time the path meets starting from . (See, for example, the vertices inside the squares in Figure 8.)
We need to prove the following.
For every , there exist two vertex-disjoint paths and between the pairs of vertices and that do not intersect the paths .
Proof of Claim: Indeed, this holds by inductively applying the combination of Lemma 4 with the assertion that for every and every with , the outgoing vertices of and and the incoming vertices of and in the layer , , , , and respectively appear in respecting the clockwise order
in the tight wall . This completes the proof of the claim.
We now construct the following paths. First, let
that is, is the union of the paths and , and is the union of the paths and . Then, for every , let
that is, is the union of the following paths; (a) the subpath of between its incoming vertex in the -th layer and its incoming vertex in the -th layer, (b) the path defined in the claim above, and (c) the subpath of between the vertices and .
Finally, for every (, if is odd) let
that is, is the union of the following three paths; (a) the subpath of between its incoming vertex in the -th layer and its incoming vertex in the -th layer, (b) the path defined in the claim above, and (c) the subpath of between the vertices and .
From the claim above and Lemma 4 we get that the above paths are vertex-disjoint. This concludes the second step of the proof.
We claim now that we may reroute the paths , , in such a way that they end up to the vertices , . Indeed, let , . From their construction these paths are vertex-disjoint and end up to the vertices , . (For a rough estimation of the position of the paths in the wall see Figure 9.)
Concluding the proof, as the mutual distance of the vertices of in the underlying non-subdivided wall is at least 2, it is easy to notice that in the annulus defined by and there exist vertex-disjoint paths from the vertices , , to the vertices of . ∎
We now prove the main result of this section.
Let be a positive integer and be a -edge-connected multigraph embedded in a surface of Eüler genus that contains a subdivided wall of height at least as a subgraph, whose compass is embedded in a closed disk . Let also be a set of vertices in the perimeter of whose mutual distance in the underlying non-subdivided wall is at least 2. If then there exist a vertex in and edge-disjoint paths from to the vertices of .
Let and be vertices belonging to the closed disk defined by the layer and to the perimeter of the wall respectively. As is -edge-connected there exist edge-disjoint paths connecting and . By Lemma 3, we may assume that the paths are confluent. Let be the family of paths , , that is, let be the family of paths consisting of he subpaths of , , between and the first vertex on which they meet the perimeter of .
Let be the set of vertices in that are contained in more than one path in . We obtain the graph by replacing every vertex with the detachment tree of in . From Observation 2, contains a wall of height whose compass is embedded in . Notice also that, as no changes have occurred in the perimeter of , and share the same perimeter. Furthermore, contains internally vertex-disjoint paths from to the perimeter of . Thus, from Lemma 5, contains vertex-disjoint paths from to . It is now easy to see, by contracting each one of the trees , , to a single vertex that contains edge-disjoint paths from to . ∎
4 Main Theorem
There exists a computable function such that for every multigraph of Eüler genus and every connected graph one of the following holds:
, where and
is not -edge-connected,
and assume that and is -edge-connected. From Lemma 1, we obtain that contains as a subgraph a subdivided wall of height whose compass is embedded in a closed disk.
In what follows we will construct a model of into the wall. From Lemma 2, admits an a orthogonal drawing in an
where the box size of each vertex is .
Notice now that can be scaled to an orthogonal drawing to the grid of size
where the box size of each vertex is , the joining vertices of each box have mutual distance at least 2 in the perimeter of the box and no joining vertex is a corner of the box.
Moreover, for every vertex , of degree , that is, for every vertex in the image of that is the intersection of two paths, there is a box in the grid of size , denoted by , containing only this vertex and vertices of the paths it belongs to. We denote by , , the vertices of belonging to the boundary of and, for uniformity, also call them joining vertices of .
Towards finding a model of in the wall observe that the grid contains as a subgraph a wall of height such that each one of the boxes, either , , or , where is the intersection of two paths in the image of contains a wall of height and the joining vertices of (the vertices , , respectively) belong to the perimeter of the wall and have distance at least 2 in it. Consider now the mapping of to where the boxes and are mapped into subwalls of of height joined together by vertex-disjoint paths as given by the orthogonal drawing . From Lemma 6, as every has height and its compass is embedded in a closed disk, there exist a vertex and edge-disjoint paths from to the joining vertices of . It is now easy to see that contains a model of . ∎
Notice now that in the case when we get the following.
There exists a computable function such that for every multigraph of Eüler genus and every connected graph one of the following holds:
is not -edge-connected,
There exists a computable function such that for every multigraph of Eüler genus