Sublinear Longest Path Transversals and Gallai Families

We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit constant-size longest path transversals. The same technique allows us to show that 2-connected graphs admit sublinear longest cycle transversals. We also make progress toward a characterization of the graphs H such that every connected H-free graph has a longest path transversal of size 1. In particular, we show that the graphs H on at most 4 vertices satisfying this property are exactly the linear forests. Finally, we show that if the order of a connected graph G is large relative to its connectivity κ(G) and α(G) ≤κ(G) + 2, then each vertex of maximum degree forms a longest path transversal of size 1.

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1 Introduction

A classic result in graph theory is that every two longest paths in a connected graph share at least one vertex. In 1966, Gallai [9] asked whether all longest paths in a connected graph have a vertex in common. The question was answered in the negative by Walther [23], who provided a counterexample with vertices. A counterexample with vertices was later constructed by Walther and Voss [24] and, independently, by Zamfirescu [27] (see Figure 1). Brinkmann and Van Cleemput (see [21]) verified that there is no counterexample with less than vertices.

Figure 1: The graph : A -vertex graph with no Gallai vertex.

A Gallai set in a graph is a set of vertices intersecting every longest path in . Equivalently, is a Gallai set in if and only if is a transversal in the hypergraph on whose edges are the vertex sets of longest paths in ; for this reason, Gallai sets are also called longest path transversals. The Gallai number of , denoted by , is the minimum size of a Gallai set and a Gallai family is a family of graphs such that for each connected graph . A vertex in is a Gallai vertex if is a Gallai set. A graph is Gallai if it has a Gallai vertex.

We are interested in two natural variants of Gallai’s question. Firstly, how large can be for a connected -vertex graph ? Secondly, which classes of graphs form Gallai families?

The graph in Figure 1 is a connected -vertex graph with . Grünbaum [12] constructed a -connected -vertex graph with . Soon afterward, Zamfirescu [27] found such a graph with vertices. Walther [23] and Zamfirescu [26] asked if is bounded for connected graphs , and this remains open. In fact, it is not known whether there is a connected graph with .

Let be a connected graph. Since does not have two vertex-disjoint longest paths, it follows that or is a Gallai set for each . Consequently, when is an -vertex connected graph. It is not too difficult to improve this argument to obtain . Rautenbach and Sereni [20] showed that for every connected -vertex graph . In Section 3, we show that when is an -vertex connected graph, implying that connected graphs have sublinear longest path transversals.

The problem of finding small Gallai sets is a special case of a general transversal problem. Given a multigraph and an edge with endpoints and , the subdivision operation produces a new multigraph in which is replaced by a path through a new vertex in . A subdivision of is a graph obtained from via a sequence of zero or more subdivision operations. In Section 3, we prove that for each connected multigraph , if the family of maximum -subdivisions in is pairwise intersecting, then admits sublinear transversals. When is the -vertex path, we obtain longest path transversals, and when is the -vertex cycle, we obtain longest cycle transversals.

Let be the smallest size of a set of vertices such that intersects every longest cycle in . Analogously to the case of longest paths in -connected graphs, every pair of longest cycles in a -connected graph intersect. The Petersen graph is -connected and . With no connectivity assumptions, Thomassen [22] showed that for each -vertex graph . The bound is sharp when is a disjoint union of triangles and nearly sharp in the -connected case when is obtained from a star with leaves by replacing each leaf with a triangle. On the other hand, Rautenbach and Sereni [20] proved that if is in addition -connected, then . In Section 3, we show that when is -connected (Corollary 2).

In the rest of the paper, we are interested in which sets of graphs form Gallai families. It is well-known that a family of pairwise intersecting subtrees of a tree has non-empty intersection; in particular, trees form a Gallai family. Several other Gallai families have been identified: split graphs and cacti [16], circular-arc graphs [1, 15], series-parallel graphs [5], graphs with matching number at most [4], dually chordal graphs [14], -free graphs [10], -sparse graphs and -free graphs [3].

Let be the class of -free graphs. A monogenic class of graphs has the form , for some graph . We would like a characterization of the monogenic Gallai families. In Section 4, we make progress by showing that if is a Gallai family, then is a linear forest, and this suffices when . In the spirit of [10], we in fact prove something more general: if is a linear forest on at most vertices and is a connected -free graph, then all maximum degree vertices in are Gallai. Dichotomies in monogenic classes for structural and algorithmic graph properties have been the subject of several studies. For example, they have been provided for properties such as boundedness of clique-width [7], price of connectivity [2, 13], and polynomial-time solvability of various algorithmic problems [11, 17, 18, 19]. In Section 5, we show that if is a connected graph with independence number (i.e., is -free), then is Gallai. We then conjecture that the same holds if .

A celebrated result of Chvátal and Erdős [6] asserts that a graph has a spanning cycle when and , and that has a spanning path when . It follows that every vertex in is Gallai when . In Section 6, we show that if a connected graph is large relative to its connectivity and , then each vertex of maximum degree is a Gallai vertex. Moreover, for each , we provide an infinite family of -connected graphs such that but no maximum degree vertex in is Gallai (see example 23).

2 Preliminaries

In this paper we consider only finite graphs. Given a graph , we denote its vertex set by and its edge set by .

Neighborhoods and degrees. For a vertex , the neighborhood is the set of vertices adjacent to in . For a set of vertices , the neighborhood of , denoted , is . We also extend the concept of neighborhood to subgraphs by defining when is a subgraph of . The degree of a vertex is the number of edges incident to in . When is clear from context, we may write for . A vertex with is cubic. The maximum degree of is . Similarly, the minimum degree of is .

Paths and cycles. A path is a non-empty graph with and . We may also denote by listing its vertices in the natural order . The vertices and are the ends or endpoints of ; the other vertices are interior vertices of . The length of is the number of edges in . We denote the -vertex path by . A -path is a path whose endpoints are and . The cycle on vertices is denoted by . The length of a cycle is the number of edges in the cycle. The girth of a graph containing a cycle is the length of a shortest cycle and a graph with no cycle has infinite girth. The distance from a vertex to a vertex in a graph is the length of a shortest path between and .

Graph operations. Let be a graph and let . The graph is obtained from by deleting all vertices in and all edges incident to a vertex in . The subgraph of induced by a set of vertices , denoted , is the graph , where . For , we define analogously. The union of simple graphs and is denoted and has vertex set and edge set . The disjoint union of and , denoted , is the union of a copy of and a copy of on disjoint vertex sets. The disjoint union of copies of is denoted by .

Graph classes and special graphs. If a graph does not contain induced subgraphs isomorphic to graphs in a set , it is -free and the set of all -free graphs is denoted by . A complete graph is a graph whose vertices are pairwise adjacent and the complete graph on vertices is denoted by . A triangle is the graph . A graph is -partite, for , if its vertex set admits a partition into classes such that every edge has its endpoints in different classes. An -partite graph in which every two vertices from distinct parts are adjacent is called complete and -partite graphs are usually called bipartite. An -bigraph is a bipartite graph with bipartition . Given a graph and , the induced -bigraph is the bipartite subgraph of with vertex set and where each edge has one endpoint in and the other in . A tree is a connected graph not containing any cycle as a subgraph and the vertices of degree are its leaves.

Graph parameters. A set of vertices or edges of a graph is maximum with respect to the property if it has maximum size among all subsets having property . An independent set of a graph is a set of pairwise non-adjacent vertices and the independence number is the size of a maximum independent set of . A clique of a graph is a set of pairwise adjacent vertices. A matching in is a set of edges with distinct endpoints. A matching saturates a set of vertices if each vertex in is the endpoint of an edge in . A graph is -connected if and is connected for each with . The connectivity of , denoted , is the maximum such that is -connected.

3 Maximum subdivision transversals

Let be a multigraph. A maximum -subdivision in is a subdivision of maximizing . An -transversal of is a set of vertices intersecting each maximum -subdivision. Let be the minimum size of an -transversal in .

Given sets of vertices and of , an -separator is a set of vertices such that no path in has one endpoint in and the other endpoint in . We allow an -separator to contain vertices in and . An -connector is a collection of vertex-disjoint paths such that each has one endpoint in , the other endpoint in , and the interior vertices of are outside . A variant of Menger’s Theorem asserts that the minimum size of an -separator equals the maximum size of an -connector (see, e.g., [8]).

Our next result shows that when the maximum -subdivisions of a graph pairwise intersect, has sublinear -transversals. We make no attempt to optimize the multiplicative constant or the dependence on .

Theorem 1.

Let be a connected -edge multigraph with and let be an -vertex graph. If the maximum -subdivisions of pairwise intersect, then .

Proof.

Let , and let . We may assume that , since otherwise we may take as our -transversal. Let be the family of maximum -subdivisions in . An -partial transversal is a triple such that is a subgraph of , , with , and each is a subgraph of or contains a vertex in . Given an -partial transversal , we either obtain an -partial transversal with or we produce an -transversal with at most vertices. Starting with and iterating gives the result.

Let be an -partial transversal, and let be the set of such that is a subgraph of . We may assume that contains vertex-disjoint paths and each of size . Otherwise, every path in has size at most , and so each has at most vertices. Since is pairwise intersecting, we have that is an -transversal for each . It follows that .

Suppose that has a -separator of size at most . Since graphs in are connected, each has a vertex in or is contained in some component of . Also, since is pairwise intersecting, at most one component of contains graphs in . Since is a separator, is disjoint from at least one of . With and , we have and . It follows that is an -partial transversal. Also since .

Otherwise, by Menger’s Theorem, has a -connector with . Let be the set of paths in of size at most . Note that , or else has at least paths of size more than , contradicting that the paths in are disjoint. Combining with two paths in whose endpoints in are as far apart as possible and a segment of gives a cycle such that , where the lower bound counts vertices in and the upper bound counts at most vertices in , at most vertices on the paths in linking and , and observing that the endpoints of the linking paths are counted twice.

Let be a longest cycle in subject to , let , and note that . If intersects each subgraph in , then witnesses . Otherwise, choose that is disjoint from . We may assume , or else witnesses that .

If has a -separator of size at most , then we obtain an -partial transversal as follows. At most one component of contains graphs in . Let and let . Since is disjoint from one of , it follows that . We compute . Hence is an -partial transversal with .

Otherwise has a -connector with . We use to obtain a contradiction. For , let be the path in corresponding to , and let be the set of paths in which have an endpoint in . Since , it follows that for some edge . Let be the set of paths in of size at most , and note that , or else has at least paths of size more than , a contradiction. The endpoints of paths in divide into edge-disjoint subpaths. Choose to minimize the length of such a subpath of , and note that has length at most ; see Figure 2. Since , we have , and hence .

Figure 2: -connector case. The subpath of the cycle is dashed, and the cycle is displayed in bold.

The endpoints of and on partition into two subpaths; let be the longer subpath. If , then we would obtain a larger -subdivision by using , , and to bypass . Since is a maximum -subdivision, we have . Therefore using , , and to bypass gives a cycle with . By the extremal choice of , it follows that . On the other hand, .

Therefore , where the last inequality uses . Simplifying gives , and this inequality is violated when . ∎

Applying Theorem 1, we obtain the following corollary.

Corollary 2.

Let be an -vertex graph. If is connected, then . If is -connected, then .

Proof.

When , an -transversal is a longest path transversal. It is well known that if is connected, then the longest paths pairwise intersect. By Theorem 1, we have .

Similarly, when , an -transversal is a longest cycle transversal. If is -connected, then the longest cycles pairwise intersect. By Theorem 1, we have . ∎

We do not know whether the assumption in Theorem 1 that is connected is necessary to obtain sublinear -transverals. To obtain analogues of Corollary 2 for general , we show that the maximum -subdivisions pairwise intersect when the connectivity of is sufficiently large.

Lemma 3.

Let be an -edge multigraph. If , then the maximum -subdivisions of are pairwise intersecting.

Proof.

Suppose for a contradiction that has disjoint maximum -subdivisions and . For each , let be the path in corresponding to . By Menger’s Theorem, there is an -connector with . For each , we associate

with an ordered pair of edges

such that has its endpoint in in and its endpoint in in . Since , some pair is associated with distinct paths . Let be the subpath of whose endpoints are in . If , then we modify to obtain a larger -subdivision by using , , and to bypass . Similarly, if , then we modify to obtain a larger -subdivision by using , , and to bypass . ∎

Corollary 4.

Let be a connected -edge multigraph. If is an -vertex graph with , then .

4 Monogenic Gallai families

In this section we make progress towards a classification of monogenic Gallai families. We first show that a necessary condition for a monogenic family to be Gallai is that is a linear forest on at most vertices, where a linear forest is a forest in which every component is a path. Let be the graph in Figure 1 with [24, 27]. We obtain necessary conditions on monogenic Gallai families by subdividing edges or replacing cubic vertices with triangles in to obtain new counterexamples with arbitrarily large girth or no induced claw, respectively.

In the following, we say that a graph is a fixer if is a Gallai family; that is, forbidding “fixes” Gallai’s conjecture.

Proposition 5.

If is a fixer, then is a linear forest on at most 9 vertices.

Proof.

Let be a fixer. By definition, if is a graph with , then is an induced subgraph of .

Note that is obtained from the Petersen graph by splitting an arbitrary vertex into a set of three vertices, each of degree 1 (see Figure 1). Clearly, is triangle-free and every path in avoids at least one vertex in . Since the Petersen graph has no spanning cycle [25], every path in omits at least vertices. Moreover, since the Petersen graph is vertex-transitive [25] and has a -cycle, it follows that for each vertex , there is a longest path in with both ends in that omits only and the other vertex in .

Let be the set of edges incident to the vertices in . Let be the graph obtained from by replacing each edge in with a path of length and replacing each edge outside with a path of length , where . Provided that , the longest paths in are in bijective correspondence with the longest paths in that have both ends in . Recalling that, for each , there is a longest path in with both ends in that omits , we have . Since has girth larger than and is an induced subgraph of , it follows that is acyclic.

Let be the set of cubic vertices in . We obtain from by replacing each vertex with a triangle such that the three edges incident to in are incident to distinct vertices of in . Clearly, is claw-free. Let be a longest path in . Again, provided that is sufficiently large, has its ends in . When visits a vertex in some , it must visit all vertices in before leaving. It follows that the longest paths in are in bijective correspondence with the longest paths in and .

Since is an induced subgraph of and , it follows that is triangle-free and claw-free, and so . Recalling that is acyclic, we have that is a linear forest. But is also an induced subgraph of and to obtain an induced linear forest as a subgraph of , a vertex must be deleted from the closed neighborhood of each cubic vertex of . Let be the set of neighbors of vertices in . Since the vertices in are cubic and have disjoint closed neighborhoods, each induced linear forest has at most vertices, and so . ∎

For , we show that is a fixer if and only if is a linear forest. Necessity follows from Proposition 5. For sufficiency, we show that every -vertex linear forest is a fixer. The linear forests of order are , , , , and . Cerioli and Lima [3] showed that -sparse graphs, a superclass of -free graphs, form a Gallai family, whereas Golan and Shan [10] showed that -free graphs form a Gallai family. In other words, and are fixers. In the following, we address the remaining cases: , , and .

Figure 3: The linear forests on vertices. These are exactly the graphs on vertices such that is a Gallai family.

We begin with some basic but useful observations. Given vertices , an -fiber is a longest path among all the -paths. Similarly, an -fiber is a longest path among all the paths having as an endpoint, and a fiber is a longest path in . Note that every fiber is an -fiber for some vertex , and every -fiber is an -fiber for some vertex .

The following two basic lemmas are used repeatedly, sometimes implicitly. Similar ideas are key to the results in [6]. The first basic lemma treats single neighbors of fibers.

Lemma 6.

Let be an -path in a graph , where with and . Let be a component of with a neighbor on . If is an -fiber, then . Moreover, if , then . Similarly, if is a -fiber, then , and if , then .

Proof.

Suppose is an -fiber. No vertex in is adjacent to , or else extends to a longer -fiber, a contradiction. Therefore, . Also, if and , then following from to , traversing , following backward from to , and traveling to produces a longer -fiber. The case that is a -fiber is symmetric. ∎

In many of our arguments, we show that a path in has some desired property or else we obtain a longer path. We now formalize two common ways to obtain longer paths. Given two lists of objects and , a splice of with is a sequence obtained from by (1) replacing a non-empty interval of with , or (2) inserting between consecutive elements in , or (3) prepending or appending to . Given a host path and a patching path , a splice of with is a path whose vertices are ordered according to a splice of the ordered list of vertices in with the ordered list of vertices in . A splice of that has the same endpoints as is an interior splice; otherwise, the splice is exterior.

A detour of an -path is a path obtained from by using two patching paths and as follows. Suppose that is a -path for and are distinct vertices appearing in order along . We follow from to , traverse , follow backward from to , traverse , and finally follow from to .

Note that our definitions of a splice and detour require the resulting object to be a path and therefore implicitly impose certain disjointness conditions on segments of the host and the patching paths. Also, note that interior splices and detours of have the same endpoints as . A splice or detour of is augmenting if it is longer than .

Let be a path in and let be a component of . A vertex with a neighbor in is an attachment point of . Our next lemma concerns pairs of attachment points.

Lemma 7.

Let be an -path in a graph and let be a component of with attachment points and , where appears before when traversing from to . The following hold.

  1. If and are consecutive on , then there is an augmenting interior splice of .

  2. If and are not consecutive along , and immediately follow and respectively, and , then there is an augmenting detour of .

  3. If and are not consecutive along , and immediately precede and respectively, and , then there is an augmenting detour of .

Proof.

For (1), since and are consecutive attachment points on , we obtain an augmenting interior splice by inserting an appropriate path in between and . For (2), let be an -path with interior vertices in and let be the path . There is an augmenting detour of using patching paths and . The case (3) is symmetric. ∎

When is a kind of fiber and a component of has may attachment points, our next lemma obtains a large independent set contained in consisting of non-attachment points.

Lemma 8.

Let be an -path in , let be a component of , let be the number of attachment points of . There is an independent set of such that , no edge joins a vertex in and a vertex in , and the following hold.

  1. If is an -fiber, then and .

  2. If is an -fiber, then and .

  3. If is a fiber, then and .

Proof.

Let be the attachment points of , with indices increasing from to along , and let .

For (1), let be the set of vertices in that immediately follow some with . Since is an -fiber, Lemma 7 implies that and are not consecutive along . Therefore, and are disjoint and so no vertex in has a neighbor in . By Lemma 7, it follows that is an independent set.

For (2), suppose in addition that is an -fiber. By Lemma 6, , and we may take to be the set of vertices that immediately follow some with .

For (3), suppose in addition that is a fiber. By Lemma 6, we have . Let be the set of vertices that immediately follow an attachment point together with . Note that since is also a -fiber, it follows from Lemma 6 that has no neighbor in , and so is an independent set of size . ∎

We can finally show in the following sections that , , and are all fixers.

4.1 is a fixer

Theorem 9.

If is a connected -free graph, then every vertex of degree at least is a Gallai vertex.

Proof.

Let be a longest path in , where with and . Suppose for a contradiction that there is a vertex with but . Let be the component of containing . Let , let be the set of attachment points of on , let , and let .

Note that is a complete graph, or else an induced copy of in together with an endpoint of would induce a copy of in . We now claim that for each . Otherwise, by Lemma 6, given a neighbor of in , would induce a copy of .

Next we claim that when . Otherwise, we obtain a longer path by starting with a neighbor of in , walking along , following from to , traversing , and following from to . Therefore for each and , otherwise would induce a copy of . It follows that for each . In particular, .

Next we claim that, if with , then . Otherwise, given a neighbor of in , the set would induce a copy of since by Lemma 7. This implies that, if , then the neighborhood of contains , , and , and so . Therefore , a contradiction. ∎

The degree assumption in Theorem 9 is best possible. Indeed, the complete bipartite graph is -free, has maximum degree , and the vertices of degree are not Gallai.

4.2 is a fixer

Proposition 10.

If is a connected -free graph, then every vertex of maximum degree is a Gallai vertex.

Proof.

Let be a connected -free graph and let be a longest path in with ends and . Suppose for a contradiction that is a vertex of maximum degree and . Let , and let be the component of containing , and let . Note that , or else we obtain a longer path by starting at a vertex in with a neighbor on and traveling around the cycle . Also, is an independent set, or else, by Lemma 6, an adjacent pair of vertices in together with and would induce a copy of .

Let be the set of attachment points of . Since has one vertex, we have . Applying Lemma 8 where is graph with the single vertex , there is an independent set such that and .

If some vertex has two non-neighbors , then induces a copy of . Hence every vertex in has at least neighbors in . Counting , every vertex in has degree at least , contradicting that . ∎

Vertices of degree in a -free graph need not be Gallai. Indeed, consider the graph obtained from by removing a matching saturating the part of size . is -free and . The longest paths in omit one vertex, and the Gallai vertices are those in the smaller part. Two of the non-Gallai vertices in the larger part have degree , which equals .

4.3 is a fixer

For a path in a graph containing the vertices and , the closed subpath of with boundary points and , denoted , is the subpath of with endpoints and . The open subpath of with boundary points and , denoted , is . Additionally, we define the semi-open subpaths and