Open Research Online: No conditions. Results ordered -Date Deposited. 2024-02-29T09:34:32ZEPrintshttps://oro.open.ac.uk/images/sitelogo.pnghttps://oro.open.ac.uk/2024-02-26T10:53:41Z2024-02-26T11:08:56Zhttps://oro.open.ac.uk/id/eprint/96123This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/961232024-02-26T10:53:41ZLower General Position Sets in GraphsA subset of vertices of a graph is a if no shortest path in contains three or more vertices of . In this paper, we generalise a problem of M. Gardner to graph theory by introducing the of , which is the number of vertices in a smallest maximal general position set of . We show that if and only if contains a universal line and determine this number for several classes of graphs, including Kneser graphs , line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.Gabriele Di StefanoSandi KlavzarAditi KrishnakumarJames Tuitejt22583Ismael Yero2024-01-22T09:54:00Z2024-02-20T10:41:50Zhttps://oro.open.ac.uk/id/eprint/95445This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/954452024-01-22T09:54:00ZThe structure of digraphs with excess oneA digraph is if for any (not necessarily distinct) vertices there is at most one directed walk from to with length not exceeding . The order of a -geodetic digraph with minimum out-degree is bounded below by the directed Moore bound . The Moore bound can be met only in the trivial cases and , so it is of interest to look for -geodetic digraphs with out-degree and smallest possible order , where is the of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for and and for and . We conjecture that there are no digraphs with excess one for and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the non-existence of certain digraphs with degree three and excess one, as well closing the open cases and left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, i.e. the outlier function of any such digraph must contain a cycle of length .James Tuitejt225832024-01-17T09:40:48Z2024-01-17T09:55:51Zhttps://oro.open.ac.uk/id/eprint/95325This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/953252024-01-17T09:40:48ZGeneral position polynomialsA subset of vertices of a graph is a general position set if no triple of vertices from the set lie on a common shortest path in . In this paper we introduce the general position polynomial as , where is the number of distinct general position sets of with cardinality . The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs , with unimodal general position polynomials are presented.Vesna IršičSandi KlavžarGregor RusJames Tuitejt225832023-09-05T15:14:55Z2023-09-05T15:30:08Zhttps://oro.open.ac.uk/id/eprint/92086This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/920862023-09-05T15:14:55ZOn some extremal position problems for graphsThe general position number of a graph is the size of the largest set of vertices such that no geodesic of contains more than two elements of . The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers. We then determine the asymptotic order of the largest size of a graph with given general or monophonic position number, classifying the extremal graphs with monophonic position number two. Finally we establish the possible diameters of graphs with given order and monophonic position number.James Tuitejt22583Elias ThomasUllas Chandran2023-03-28T10:19:19Z2023-04-06T14:47:46Zhttps://oro.open.ac.uk/id/eprint/88156This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/881562023-03-28T10:19:19ZSmall Graphs and Hypergraphs of Given Degree and GirthThe search for the smallest possible d-regular graph of girth g has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a d-regular, r-uniform hypergraph of given (Berge) girth g. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques to find new record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.Grahame Erskinege2723James Tuitejt225832023-03-24T11:21:48Z2023-04-06T11:06:46Zhttps://oro.open.ac.uk/id/eprint/88147This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/881472023-03-24T11:21:48ZOn monophonic position sets in graphsThe general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.Elias John ThomasUllas Chandran S.V.James Tuitejt22583Gabriele Di Stefano2023-02-21T15:28:35Z2024-02-14T10:06:00Zhttps://oro.open.ac.uk/id/eprint/87516This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/875162023-02-21T15:28:35ZOn the vertex position number of graphsIn this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex of a graph , we say that a set is an if for any the shortest -paths in contain no point of . We investigate the largest and smallest orders of maximum -position sets in graphs, determining these numbers for common classes of graphs and giving bounds in terms of the girth, vertex degrees, diameter and radius. Finally we discuss the complexity of finding maximum vertex position sets in graphs.Maya ThankachyUllas ChandranJames Tuitejt22583Elias ThomasGabriele Di StefanoGrahame Erskinege27232023-02-21T15:18:54Z2024-02-22T09:41:40Zhttps://oro.open.ac.uk/id/eprint/87504This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/875042023-02-21T15:18:54ZTraversing a Graph in General PositionLet G be a graph. Assume that to each vertex of a set of vertices S⊆V(G) a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited whilst maintaining the general position property at all times. The mobile general position number of G is the cardinality of a largest mobile general position set of G. In this paper, bounds on the mobile general position number are given and exact values determined for certain common classes of graphs including block graphs, rooted products, unicyclic graphs, Cartesian products, joins of graphs, Kneser graphs K(n,2) and line graphs of complete graphs.Sandi KlavzarAditi KrishnakumarJames Tuitejt22583Ismael Yero2023-02-21T12:41:23Z2023-03-02T11:53:29Zhttps://oro.open.ac.uk/id/eprint/87574This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/875742023-02-21T12:41:23ZJisc_PRTurán Problems for <i>k</i> -Geodetic DigraphsA digraph G is k-geodetic if for any pair of (not necessarily distinct) vertices u, v∈V(G) there is at most one walk of length ≤k from u to v in G. In this paper, we determine the largest possible size of a k-geodetic digraph with a given order. We then consider the more difficult problem of the largest size of a strongly-connected k-geodetic digraph with a given order, solving this problem for k=2 and giving a construction which we conjecture to be extremal for larger k. We close with some results on generalised Turán problems for the number of directed cycles and paths in k-geodetic digraphs.James Tuitejt22583Grahame Erskinege2723Nika Salia2022-09-20T10:36:03Z2023-03-24T21:36:10Zhttps://oro.open.ac.uk/id/eprint/85084This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/850842022-09-20T10:36:03ZOn Networks with Order Close to the Moore BoundThe degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the degree/geodecity problem concerns the smallest order of a k-geodetic mixed graph with given minimum undirected and directed degrees; this is a generalisation of the classical degree/girth problem. In this paper we present new bounds on the order of mixed graphs with given diameter or geodetic girth and exhibit new examples of directed and mixed geodetic cages. In particular, we show that any k-geodetic mixed graph with excess one must have geodetic girth two and be totally regular, thereby proving an earlier conjecture of the authors.James Tuitejt22583Grahame Erskinege27232022-06-06T15:44:34Z2023-03-30T03:25:42Zhttps://oro.open.ac.uk/id/eprint/82741This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/827412022-06-06T15:44:34ZExtremal Directed And Mixed GraphsWe consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Turn problems, in the context of directed and partially directed graphs.

A directed graph or mixed graph is -geodetic if there is no pair of vertices of such that there exist distinct non-backtracking walks with length in from to . The order of a -geodetic digraph with minimum out-degree is bounded below by the ; similarly the order of a -geodetic mixed graph with minimum undirected degree and minimum directed out-degree is bounded below by the . We will be interested in networks with order exceeding the Moore bound by some small .

The asks for the smallest possible order of a -geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call , and provide some bounds on their order and information on their structure.

We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages.

We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of Lpez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular.

Using counting arguments we then provide strong bounds on the order of totally regular -geodetic mixed graphs and use these results to derive new extremal mixed graphs.

Finally we change our focus and study the Turn problem of the largest possible size of a -geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected -geodetic digraphs, as well as providing constructions of strongly connected -geodetic digraphs that we conjecture to be extremal for larger . We close with a discussion of some related generalised Turn problems for -geodetic digraphs.James Tuitejt225832022-03-14T14:19:58Z2023-03-24T10:21:23Zhttps://oro.open.ac.uk/id/eprint/82211This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/822112022-03-14T14:19:58ZSome Position Problems for GraphsThe general position problem for graphs stems from a puzzle of Dudeney and the general position problem from discrete geometry. The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but with ‘induced path’ in place of ‘geodesic’. In this abstract we discuss the smallest possible order of a graph with given general and monophonic position numbers, determine the asymptotic order of the largest size of a graph with given order and position numbers and finally determine the possible diameters of a graph with given order and monophonic position number.James Tuitejt22583Elias John ThomasUllas Chandran S. V.2021-09-24T14:34:17Z2023-03-24T10:20:31Zhttps://oro.open.ac.uk/id/eprint/79106This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/791062021-09-24T14:34:17ZNew Bounds on <i>k</i>-Geodetic Digraphs and Mixed GraphsWe study a generalisation of the degree/girth problem to the setting of directed and mixed graphs. We say that a mixed graph or digraph G is k-geodetic if there is no pair of vertices u, v such that G contains distinct non-backtracking walks of length ≤ k from u to v. The order of a k-geodetic mixed graph with minimum undirected degree r and minimum directed out-degree z in general exceeds the mixed Moore bound M(r, z, k) by some small excess ϵ. Bannai and Ito proved that there are no non-trivial undirected graphs with excess one. In this paper we investigate the structure of digraphs with excess one and derive results on the permutation structure of the outlier function that rules out the existence of certain digraphs with excess one. We also present strong bounds on the excess of k-geodetic mixed graphs and show that there are no k-geodetic mixed graphs with excess one for k ≥ 3 .James Tuitejt22583Grahame Erskinege27232019-10-30T11:41:13Z2023-06-30T21:27:20Zhttps://oro.open.ac.uk/id/eprint/67892This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/678922019-10-30T11:41:13ZOn Total Regularity of Mixed Graphs with Order Close to the Moore BoundThe undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter , maximum undirected degree and maximum directed out-degree . Similarly one can search for the smallest possible -geodetic mixed graphs with minimum undirected degree and minimum directed out-degree . A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For , we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of Lopez and Miret. We also present partial results for larger . We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.James Tuitejt22583Grahame Erskinege27232019-09-17T12:10:10Z2023-03-24T10:20:12Zhttps://oro.open.ac.uk/id/eprint/66734This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/667342019-09-17T12:10:10ZOn diregular digraphs with degree two and excess threeMoore digraphs, that is digraphs with out-degree d, diameter k and order equal to the Moore bound M(d, k) = 1 + d + d^{2} + · · · + d^{k}, arise in the study of optimal network topologies. In an attempt to find digraphs with a ‘Moore-like’ structure, attention has recently been devoted to the study of small digraphs with minimum out-degree d such that between any pair of vertices u, v there is at most one directed path of length ≤ k from u to v; such a digraph has order M(d, k)+ϵ for some small excess ϵ. Sillasen et al.
have shown that there are no digraphs with minimum out-degree two and excess one (Miller et al., 2018; Sillasen, 2015). The present author has classified all digraphs with out-degree two and excess two (Tuite, 2016, 2018). In this paper it is proven that there are no diregular digraphs with out-degree two and excess three for k ≥ 3, thereby providing the first classification of digraphs with order three away from the Moore bound for a fixed out-degree.James Tuitejt225832019-01-30T14:07:00Z2023-03-25T01:25:31Zhttps://oro.open.ac.uk/id/eprint/58923This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/589232019-01-30T14:07:00ZDigraphs with degree two and excess two are diregularA k-geodetic digraph with minimum out-degree d has excess ϵ if it has order M(d,k)+ϵ, where M(d,k) represents the Moore bound for out-degree d and diameter k. For given ϵ, it is simple to show that any such digraph must be out-regular with degree d for sufficiently large d and k. However, proving in-regularity is in general non-trivial. It has recently been shown that any digraph with excess ϵ=1 must be diregular. In this paper we prove that digraphs with minimum out-degree d=2 and excess ϵ=2 are diregular for k≥2.James Tuitejt225832018-07-23T13:27:46Z2023-03-24T12:38:06Zhttps://oro.open.ac.uk/id/eprint/55855This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/558552018-07-23T13:27:46ZLarge Cayley graphs of small diameterThe degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we restrict attention to Cayley graphs, and study the asymptotics by fixing a small diameter and constructing families of graphs of large order for all values of the maximum degree. Much of the literature in this direction is focused on the diameter two case. In this paper we consider larger diameters, and use a variety of techniques to derive new best asymptotic constructions for diameters 3, 4 and 5 as well as an improvement to the general bound for all odd diameters. Our diameter 3 construction is, as far as we know, the first to employ matrix groups over finite fields in the degree-diameter problem.Grahame Erskinege2723James Tuitejt225832018-01-26T10:17:32Z2023-06-30T19:06:00Zhttps://oro.open.ac.uk/id/eprint/52980This item is in the repository with the URL: https://oro.open.ac.uk/id/eprint/529802018-01-26T10:17:32ZOn diregular digraphs with degree two and excess twoAn important topic in the design of efficient networks is the construction of (d, k, +Є)- digraphs, i.e. k-geodetic digraphs with minimum out-degree ≥ d and order M(d,k)+ Є, where M(d,k) represents the Moore bound for degree d and diameter k and Є > 0 is the (small) excess of the digraph. Previous work has shown that there are no (2, k,+1)-digraphs for k ≥ 2. In a separate paper, the present author has shown that any (2, k,+2)-digraph must be diregular for k ≥ 2. In the present work, this analysis is completed by proving the nonexistence of diregular (2, k,+2)-digraphs for k ≥ 3 and classifying diregular (2,2,+2)-digraphs up to isomorphism.James Tuitejt22583