1 Introduction
Computing the diameter of a graph (maximum number of edges on a shortest path) is a fundamental problem with countless applications in computer science and beyond. Unfortunately, the textbook algorithm for computing the diameter of an vertex edge graph takes time. This quadratic runningtime is too prohibitive for large graphs with millions of nodes. As already noticed in [15, 16], an algorithm breaking this quadratic barrier for general graphs is unlikely to exist since it would lead to more efficient algorithms for some disjoint set problems and, as proved in [30], the latter would falsify the Strong ExponentialTime Hypothesis (SETH). This raises the question of when we can compute the diameter faster than . By restricting ourselves to more structured graph classes, here we hope in obtaining a finergrained dichotomy between “easy” and “hard” instances for diameter computations – with the former being quasi lineartime solvable and the latter being impossible to solve in subquadratic time under some complexity assumptions.
Specifically, we focus in this work on the class of split graphs, i.e., the graphs that can be bipartitioned into a clique and a stable set. – For any undefined graph terminology, see [5]. – This is one of the most basic idealized models of core/periphery structure in complex networks [7]. We stress that every split graph has diameter at most three. In particular, computing the diameter of a noncomplete split graph boils down to decide whether this is either two or three. Nevertheless, under the Strong ExponentialTime Hypothesis (SETH) the textbook algorithm is optimal even for split graphs [6]. We observe that the split graphs are in some sense the hardest instances for deciding whether the diameter is at most two. For that, let us bipartition a split graph into a maximal clique and a stable set ; it takes linear time [23]. The sparse representation of is defined as ^{4}^{4}4 This sparse representation is also called the neighbourhood set system of the stable set of , see Sec. 2.. – Note that all our algorithms in this paper run in time linear in the size of this above representation. –
Observation 1
Deciding whether an vertex graph has diameter two can be reduced in linear time to deciding whether a vertex split graph (given by its sparse representation) has diameter two.
Proof. If then, let be a disjoint copy of . For every we denote by its copy. We define where . By construction, is a split graph with maximal clique and stable set . Furthermore, if and only if .
We here address the finegrained complexity of diameter computation on subclasses of split graphs. By the above Observation 1, our results can be applied to the study of the diametertwo problem on general graphs.
Related work.
Exact and approximate distance computations for chordal graphs: a farreaching generalization of split graphs, have been a common research topic over the last few decades [10, 11, 17, 18]. To the best of our knowledge, this is the first study on the complexity of diameter computation on split graphs. However, there exist lineartime algorithms for computing the diameter on some other subclasses of chordal graphs such as: interval graphs [29], or strongly chordal graphs [9]. These results imply the existence of lineartime algorithms for diameter computations on interval split graphs and strongly chordal split graphs, among other subclasses.
Beyond chordal graphs, the complexity of diameter computation has been considered for many graph classes, e.g., see [16] and the papers cited therein. In particular, the diameter of general graphs with treewidth can be computed in quasi linear time [1], whereas under SETH we cannot compute the diameter of split graphs with cliquenumber (and so, treewidth) in subquadratic time [6]. We stress that our two first examples of “easy” subclasses, namely: interval split graphs and strongly chordal split graphs have unbounded treewidth. Our work unifies almost all known tractable cases for diameter computation on split graphs – and offers some new such cases – through a new increasing hierarchy of subclasses.
Relatedly, we proved in a companion paper [20] that on the proper minorclosed graph classes and the boundeddiameter graphs of constant distance VCdimension (not necessarily split) we can compute the diameter in time , for some small . We consider in this work some subclasses of split graphs of constant distance VCdimension. However, the time bounds obtained in [20] are barely subquadratic. For instance, although the graphs of constant treewidth fit in our framework, our techniques in [20] do not suffice for computing their diameter in quasi linear time. In fact, neither are they sufficient to explain why we can compute the diameter in subquadratic time on graphs of superconstant treewidth . Unlike [20] this article is a new step toward characterizing the graph classes for which we can compute the diameter in quasi linear time.
Our results.
We introduce a new invariant for split graphs, that we call the cliqueinterval number. Formally, for any , a split graph is cliqueinterval if the vertices in the clique can be totally ordered in such a way that the neighbours of any vertex in the stable set form at most intervals. The cliqueinterval number of a split graph is the minimum such that it is cliqueinterval. Although this definition is quite similar to the one of the interval number [28], we show in Sec. 2 that being cliqueinterval does not imply being interval, and viceversa^{5}^{5}5 We observe that the general graphs with bounded interval number are sometimes called “split interval” [2], that may create some confusion.. In fact, as we also proved in Sec. 2, the cliqueinterval number of a split graph is more closely related to the VCdimension and the stabbing number of the neighbourhood set system of its stable set. Nevertheless, a weak relationship with the interval number can also be derived in this way.
We then study in Sec. 3 what the complexity of computing the diameter is on split graphs parameterized by the cliqueinterval number. It follows from our results in Sec. 2 and those in [20] that on every subclass of constant cliqueinterval number, there exists a subquadratictime algorithm for diameter computation. Our work completes this general result as it provides an almost complete characterization of the quasi lineartime solvable instances. As a warmup, we observe in Sec. 3.1 that on cliqueinterval split graphs (a.k.a., cliqueinterval), deciding whether the diameter is two is equivalent to testing for a universal vertex. We give a direct proof of this result and another one based on the inclusion of cliqueinterval split graphs in the subclass of strongly chordal split graphs. On the way, we prove more generally – and perhaps surprisingly – that for the intersection of split graphs with many interesting graph classes from the literature, having diameter at most two is equivalent to having a universal vertex! Then, we address the more general case of cliqueinterval split graphs, for .

Our first main contribution is the following almost dichotomy result (Theorem 2). For every vertex cliqueinterval split graph, we can compute its diameter in quasi lineartime if and a corresponding total ordering of its clique is given. This result follows from an all new application of a generic framework based on range trees [3], that was already used for diameter computations on some special cases [1, 19] but with a quite different approach than ours. Furthermore, the logarithmic upper bound on is somewhat tight. Indeed, we also prove that under SETH we cannot compute in subquadratic time the diameter of cliqueinterval split graphs for .

Then, we focus on the complements of cliqueinterval split graphs — that are an interesting subclass in their own right since they generalize, e.g., interval split graphs. For the latter we get a more straightforward algorithm for diameter computation, with a better dependency in . Indeed, we prove that we can compute the diameter of such graphs in time if a corresponding ordering is given. This result is conditionally optimal up to polylogarithmic factors because and, under SETH, we cannot compute the diameter of split graphs with edges in subquadratic time [1].
It follows from these two above results that having at hands a cliqueinterval ordering for a split graph or its complement can help to significantly improve the time complexity for computing its diameter. We so ask whether such orderings can be computed in quasi linear time, for some small values of . In Sec. 4 we prove that it is indeed the case for boundedtreewidth split graphs (trivially) and some other dense subclasses of bounded cliqueinterval number such as comparability split graphs. Finally our main result in this section is that the cliqueinterval split graphs can be recognized in linear time.
Overall, we believe that our study of cliqueinterval split graphs is a promising framework in order to prove, somewhat automatically, new quasi lineartime solvable special cases for diameter computations on split graphs and beyond.
2 Cliqueinterval numbers and other graph parameters
We start by relating the cliqueinterval number of split graphs with betterstudied invariants from Graph theory and Computational geometry.
Treewidth.
First we observe that if is a split graph then, for any total order over and any , is the union of at most intervals. In fact we can improve this rough upperbound, as follows:
Lemma 1
Every split graph with cliquenumber (and so, treewidth) at most is cliqueinterval.
Proof. Let be a split graph with cliquenumber . Let be any total ordering of . For any integer , and any , we claim that is the union of at most intervals of . Indeed if has at least nonneighbours, then it has at most neighbours and so the claim trivially holds. Otherwise, has less than nonneighbours in , but then the other nodes form at most intervals of . This bound is optimal for .
Conversely, since any complete graph is cliqueinterval, the cliqueinterval number cannot be bounded by any function of the treewidth.
VCdimension and Stabbing number.
For a set system (a.k.a., range space or hypergraph), we say that a subset is shattered if is the powerset of . The VCdimension of a finite set system is the largest size of a shattered subset. We now prove an intriguing connection between the cliqueinterval number of a split graph and the VCdimension of a related set system.
Proposition 1
For any split graph , let . If is cliqueinterval then has VCdimension at most .
Proof. Suppose by contradiction that is cliqueinterval but has VCdimension at least . Let be such that and is shattered. Any total ordering of induces a total ordering over . Since is shattered there is an such that . But then, is the union of at least intervals, a contradiction.
It turns our that a week converse of Proposition 1 also holds. We need a bit of terminology from [14]. A spanning path for is a total ordering of . Its stabbing number is the maximum number of consecutive pairs that a set can stab, i.e., for which . Finally, the stabbing number of is the minimum stabbing number over its spanning paths.
Observation 2
For any split graph , let . The cliqueinterval number of is up to one the stabbing number of .
The main result of [14] is that every range system of VCdimension at most has a stabbing number in , for some exponential function . We so obtain:
Corollary 1
For any split graph , let . If has VCdimension at most then is cliqueinterval, for some exponential function .
Interval number.
Finally, we relate the cliqueinterval number of split graphs with their interval number. A graph is called interval if we can map every to the union of at most closed interval on the real line, denoted by , in such a way that . In particular, interval graphs are exactly the interval graphs.
We observe that there are interval split graphs that are not cliqueinterval, already for . For instance, consider the interval split graph of Fig. 1 (note that is a linear ordering of its maximal cliques). Suppose by contradiction it is cliqueinterval, and let us consider a corresponding total ordering of . The intersection should be an interval. But then, one of the pairs or is not consecutive, and so one of or is not an interval. Therefore, the graph of Fig. 1 is not cliqueinterval. Conversely, there are cliqueinterval split graphs which are not interval graphs. For instance, this is the case of thin spiders, i.e., split graphs such that the edges between the maximal clique and the stable set induce a perfect matching.
Nevertheless we prove a weak connection between the cliqueinterval number and the interval number of a split graph, by using the VCdimension. Specifically, Bousquet et al. proved that the neighbourhood set system of any interval graph has VCdimension at most two [8]. We generalize their result.
Proposition 2
The neighbourhood set system of any interval graph has VCdimension at most .
Proof. Let be a interval graph, and let us fix a corresponding interval representation. We create a graph whose vertices are the intervals in this representation and such that there is an edge between every two intersecting intervals. Observe that is an interval graph, and so, its neighbourhood system has VCdimension at most two [8]. Now, let , , be shattered by the neighbourhood set system of . For the corresponding interval set , the cardinality of is an by the SauerShelahPerles’ Lemma [31, 33]. But then, for any , there are only possibilities for . Since is shattered and, for any , , we so obtain . As a result, we must have that .
Corollary 2
If is a interval split graph then, is also cliqueinterval, for some exponential function .
3 Diameter Computation in quasi linear time
We now address the time complexity of diameter computation on cliqueinterval split graphs. Our first result in this section follows from the relations proved in Sec. 2 with the VCdimension.
Theorem 1 ( [20])
For every , there exists a constant such that in time we can decide whether a graph whose neighbourhood set system has VCdimension at most has diameter two.
Corollary 3
For every constant , there exists a constant such that in time we can decide whether a cliqueinterval split graph has diameter two.
Proof. Let be a cliqueinterval split graph. By Theorem 1, it suffices to prove that the neighbourhood set system of has VCdimension bounded by a function of . The latter system is the union of with . Since the VCdimension of the union of two set systems can be upper bounded by a function of their respective VCdimensions [21], we are left proving that both set systems have a VCdimension which is upper bounded by a function of . By Proposition 1, has VCdimension at most . Furthermore, since we have for every , the VCdimension of is up to some constant the same as for . We observe that and are dual from each other. Therefore, the VCdimension of is at most an exponential of the VCdimension of [14]. We conclude from Proposition 1 that the VCdimension of is at most , and we are done using Theorem 1.
We observe that Corollary 3 also holds for the complements of cliqueinterval split graphs – with essentially the same proof as above. In the remainder of this section we focus on the existence of quasi lineartime algorithms. It starts with a small digression on cliqueinterval split graphs.
3.1 The Case and beyond
Proposition 3
A cliqueinterval split graph has diameter at most two if and only if it has a universal vertex. In particular, we can compute the diameter of cliqueinterval split graphs in linear time.
Proof. For cliqueinterval, let us fix a corresponding total order over the maximal clique . Since is a dominating clique of , all vertices in are at a distance at most two from any other vertex in . Therefore, we are left to decide whether every two vertices in the stable set are at distance two, i.e., whether they have a common neighbour. Equivalently, given the total order over , we must decide whether for every , the intervals and intersect. By the Helly property, a finite family of intervals on the line pairwise intersect if and only if they have a nonempty intersection. As a result, we are left deciding whether , or equivalently whether some vertex of is adjacent to all of . Since is a clique, this is equivalent to having a universal vertex.
We think that our proof of Prop. 3 is a nice introduction to the properties of cliqueinterval representations for split graphs, that will be further exploited in Sec. 3.2. In the remainder of this part we give an alternative proof of Prop. 3 that also applies to larger subclasses of split graphs. It starts with the following inclusion lemma.
Lemma 2
Every cliqueinterval split graph is strongly chordal.
Proof. The sun graph is a split graph such that and for every we have (indices are taken modulo ). A chordal graph is strongly chordal if and only if it does not contain any sun graph as an induced subgraph, for every [22]. This is always the case for cliqueinterval split graphs because, in any sun graph, the neighbourhoods of the vertices in the stable set induce a cyclic ordering over .
We stress that since every sun graph is cliqueinterval, this above Lemma 2 does not hold for cliqueinterval graphs, for any . Now, a maximum neighbour of is any such that (see [9]). We prove that if at least one vertex in a graph has a maximum neighbour then deciding whether the diameter is at most two becomes a trivial task. In particular, this is always the case for strongly chordal graphs [22].
Lemma 3
For every and such that is a maximum neighbour of , we have if and only if is a universal vertex.
Proof. If is universal in then, trivially, . Conversely, assume . Then, for every , , and so there exists a such that . In particular, . As a result, we have or, equivalently, is universal.
Prop. 3 now follows from the combination of Lemmata 2 and 3. Furthermore it turns out that many wellstructured graph classes ensure the existence of a vertex with a maximum neighbour such as: graphs with a pendant vertex, threshold graphs [25] and interval graphs [12], or even more generally dually chordal graphs [9]. We conclude that:
Corollary 4
We can compute the diameter of split graphs with minimum degree one and dually chordal split graphs in linear time.
In particular, we can compute the diameter of interval split graphs and strongly chordal split graphs in linear time.
We observe that we can easily modify the hardness reduction from [6] in order to show that, under SETH, we cannot compute in subquadratic time the diameter of split graphs with minimum degree two. Indeed, this aforementioned reduction outputs a split graph such that: is partitioned in two disjoint subsets and ; and a diametral pair must have one end in each subset. Let be fresh new vertices, and let be such that: and:
By construction, if and only if . Therefore, Corollary 4 is optimal for the parameter minimum degree.
3.2 The general case
We are now ready to prove the first main result of this paper.
Theorem 2
If is an vertex cliqueinterval split graph and a corresponding total order over is given then, we can compute the diameter of in time . This is quasi lineartime if .
Conversely, under SETH we cannot compute the diameter of vertex split graphs with cliqueinterval number in subquadratic time.
In order to prove Theorem 2, we will use a special instance of range tree. Such a datastructure stores a static set of dimensional points and it supports the following operation:

(Range Query) Given intervals , compute the number of stored points such that, for every , we have ^{6}^{6}6 We refer to [13] for a more general presentation of range trees..
Proposition 4 ([13])
For any and any set of dimensional points, we can construct a range tree in time and answer any range query in time .
The use of range queries for diameter computation dates back from [1] (see also [13, 19] for some further applications). Roughly, if in a graph we can find a separator of size at most that disconnects a diametral pair of , then the idea is to compute a “distance profile” for every vertex w.r.t. , and to see this profile as a dimensional point. We can compute a diametral pair by constructing a range tree for these points and computing range queries. Our present approach is different from the one in [1] as we define our multidimensional points based on some interval representation of the graph rather than on distance profiles, and we use a different type of range query than in [1].
Proof of Theorem 2. Let be a cliqueinterval split graph, and a corresponding total order over . By the hypothesis for every , we have is the union of intervals such that . Furthermore since the ordering over is given, the endpoints that delimit these intervals can be computed in time , simply by scanning the neighbours of vertex . We so map every vertex to the dimensional point , that takes total time . Then, we construct a range tree for the points , that takes time by Proposition 4.
For every , we are left with computing the number of vertices in at distance two from . Indeed, has diameter at most two if and only if this number is for every . More specifically, for every we want to compute the number of vertices such that:

;

while for every , ;

and for every and , .
The first and second constraints are equivalent to one of the following three disjoint possibilities:

and ;

or and ;

or .
The third constraint is equivalent to have . Note that in order to subdivide this constraint into disjoint possibilities, it suffices to indicate: (i) the subset of all intervals among that contain an interval for some ; and (ii) the set of all indices such that and are not contained in the same such interval. Overall, that divides the third constraint in at most disjoint events. For a fixed pair we so reduce our computation to at most range queries, that takes time by Proposition 4. Since there are such pairs, the total time in order to compute the number of vertices in at distance two from is an .
Finally, the hardness result for follows from the fact that treewidth split graphs are cliqueinterval (Lemma 1) and that under SETH, we cannot compute the diameter of split graphs with treewidth in subquadratic time [1, 6].
Before ending this section, we give a simpler algorithm for computing the diameter on the complements of cliqueinterval split graphs. It is similar in spirit to [20, Lemma 6].
Theorem 3
If is the complement of a cliqueinterval split graph and a corresponding total order over is given then, we can compute the diameter of in time .
Proof. By the hypothesis for every , we have is the union of intervals. In particular, is the union of intervals such that . Furthermore since the ordering over is given, the endpoints that delimit these intervals can be computed in time , simply by scanning the neighbours of vertex . Overall, this preprocessing phase takes total time . Then in order to compute the diameter of , for every we store the endpoints for every . This takes time , and so total time . Furthermore a vertex has eccentricity at most two if and only if we have , that is equivalent to have . In order to check whether this collection of intervals covers all of , it suffices to sort the pairs for all and
, and then to scan these ordered pairs from left to right. This can be done in time
. As a result, we can decide whether in total time .4 Recognition of Cliqueinterval Split graphs
Our two algorithms in Sec. 3.2 show that in order to compute the diameter of cliqueinterval split graphs in quasi linear time, it is sufficient to compute a corresponding total order of their maximal clique. This raises the question whether such cliqueinterval orderings can always be computed in quasi linear time. A first positive example was given by Lemma 1. Indeed, for a split graph of treewidth at most , we can pick any total order of its maximal clique. We complete this easy result by Sec. 4.1 where we give examples of dense subclasses of split graphs with constant cliqueinterval number and for which a corresponding order can be computed in linear time. Finally, in Sec. 4.2 we prove a stronger result, namely that we can recognize the cliqueinterval graphs in linear time.
4.1 Examples of subclasses with bounded cliqueinterval number
A threshold graph is a split graph such that: (i) the neighbourhoods of the vertices in and (ii) the neighbourhoods of the vertices in are totally ordered by inclusion. Observe that threshold graphs can be dense and of unbounded treewidth.
Lemma 4
Every threshold graph is cliqueinterval.
Proof. For a threshold graph let be such that . In order to prove that is cliqueinterval, it suffices to construct any total order of such that the subsets are consecutive intervals.
Note that we can easily derive from the proof of Lemma 4 a lineartime algorithm for computing a cliqueinterval ordering.
Finally, a comparability graph is a graph that admits a transitive orientation.
Lemma 5
For every comparability split graph , we can compute in linear time a total order over such that, for every , is the union of a prefix and a suffix of this order.
In particular, every comparability split graph is cliqueinterval.
Proof. A comparability ordering of is a total order over with the property that, for every , [26]. For a given comparability graph , we can compute a comparability ordering in linear time [27]. Then, let be the subordering induced by over . For every , we claim that is a (possibly empty) prefix of . Indeed, suppose by contradiction there exist such that . Since we should have , a contradiction. Therefore, the claim is proved. We can prove similarly that is a (possibly empty) suffix of .
We also want to stress that the complements of comparability split graphs, i.e., the cocomparability split graphs are just interval split graphs and we have already considered this case in Section 3.1.
Remark 1. The ordering given by Lemma 5 has some additional properties that can be used for computing the diameter of comparability split graphs in linear time. Indeed, for every such that and are nonempty we have either or because these are prefixes of the ordering over . In particular, and are at distance two from each other. The same holds if both and are nonempty. Hence, let and . We are left deciding whether every are pairwise at distance two. For that, since all the sets are prefixes of the total order over , and in the same way all the sets are suffixes of this ordering, we only need to consider a pair such that and are minimized.
4.2 Lineartime recognition of Cliqueinterval graphs
Theorem 4
Cliqueinterval split graphs can be recognized in linear time.
Proof. Let be a split graph. We define a graph from by first transforming into a stable set and then, for every , making a clique of . Furthermore, we claim that is cliqueinterval if and only if is interval. To see that, let us call cliquepath of a graph an ordering of its maximal cliques such that, for any vertex of , the maximal cliques containing are contiguous in the ordering; a graph is interval if and only if it admits a cliquepath [4]. We can now prove our claim, as follows:

If is cliqueinterval then, any cliqueordering over for is a total ordering over the maximal cliques , for . Furthermore as already observed in the proof of Proposition 3 the vertices in a subset are pairwise at distance two in if and only if they have a common neighbour in . As a result, the maximal cliques of are exactly the sets , and so admits a cliquepath. This implies that is an interval graph.

Conversely, if is an interval graph then any cliquepath of induces a total ordering over the maximal cliques , and so, a cliqueordering over for .
Unfortunately, computing the graph may take superlinear time. We can overcome this issue as follows. First, given a family of subsets over , if there exists a chordal graph whose maximal cliques are exactly those in then, we can compute a LexBFS ordering of in time (Algorithm 10 in [24]). Moreover we can also compute a cliquetree of in time [34, Sec. 3]. Finally given this LexBFS ordering and the corresponding cliquetree of , we can apply Algorithm 9 from [24] in order to decide in time whether is interval. For solving our initial problem, we can take , and in this situation we have .
We left open the status of the recognition of cliqueinterval split graphs, for .
5 Open problems
Although the definitions of cliqueinterval and interval split graphs have some similarities, we observe that computing the diameter of interval split graphs in quasi linear time already looks like a challenging task. Indeed, for a interval split graph and , the vertices at distance two from are exactly those such that one of their intervals intersects one of the intervals that represent the neighbours of . We cannot use our range query framework in order to avoid overcounting these vertices as this would require up to range queries. More generally, for every fixed , can we compute the diameter of interval graphs in quasi linear time? We stress that every planar graph is interval [32], and that the complexity of diameter computation on this class of graphs is a longstanding open problem. The case could thus be an interesting intermediate step.
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