The forcing hull and forcing geodetic numbers of graphs

AbstractFor every pair of vertices u,v in a graph, a u–v geodesic is a shortest path from u to v. For a graph G, let IG[u,v] denote the set of all vertices lying on a u–v geodesic. Let S⊆V(G) and IG[S] denote the union of all IG[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if IG[S]=S. A convex hull [S]G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if IG[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number fh(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number fg(G) of G. In the paper, we construct some 2-connected graph G with (fh(G),fg(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (fh(G),fg(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81–94]

On the Forcing Hull and Forcing Monophonic Hull Numbers of Graphs

For a connected graph G = (V,E), let a set M be a minimum monophonic hull set of G. A subset T ⊆ M is called a forcing subset for M if M is the unique minimum monophonic hull set containing T

On the hull and interval numbers of oriented graphs

In this work, for a given oriented graph $D$, we study its interval and hull numbers, denoted by ${in}(D)$ and ${hn}(D)$, respectively, in the geodetic, ${P_3}$ and ${P_3^*}$ convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph $D$, we prove that ${hn_g}(D)\leq m(D)-n(D)+2$ and that there is a strongly oriented graph such that ${hn_g}(D) = m(D)-n(D)$. We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allows us to deduce polynomial-time algorithms to compute ${hn_{P_3}}(D)$ when the underlying graph of $D$ is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ${in_g}(D)\leq k$ or ${hn_g}(D)\leq k$ is NP-hard or W[i]-hard parameterized by $k$, for some $i\in\mathbb{Z_+^*}$, then the same holds even if the underlying graph of $D$ is bipartite. Next, we prove that deciding whether ${hn_{P_3}}(D)\leq k$ or ${hn_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is bipartite; that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is NP-complete, even if $D$ has no directed cycles and the underlying graph of $D$ is a chordal bipartite graph; and that deciding whether ${in_{P_3}}(D)\leq k$ or ${in_{P_3^*}}(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is split. We also argue that the interval and hull numbers in the oriented $P_3$ and $P_3^*$ convexities can be computed in polynomial time for graphs of bounded tree-width by using Courcelle's theorem

Symbolic regression for approximating graph geodetic number

Graph properties are certain attributes that could make the structure of the graph understandable. Occasionally, standard methods cannot work properly for calculating exact values of graph properties due to their huge computational complexity, especially for real-world graphs. In contrast, heuristics and metaheuristics are alternatives proved their ability to provide sufficient solutions in a reasonable time. Although in some cases, even heuristics are not efficient enough, where they need some not easily obtainable global information of the graph. The problem thus should be dealt in completely different way by trying to find features that related to the property and based on these data build a formula which can approximate the graph property. In this work, symbolic regression with an evolutionary algorithm called Cartesian Genetic Programming has been used to derive formulas capable to approximate the graph geodetic number which measures the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Finding the exact value of the geodetic number is known to be NP-hard for general graphs. The obtained formulas are tested on random and real-world graphs. It is demonstrated how various graph properties as training data can lead to diverse formulas with different accuracy. It is also investigated which training data are really related to each property

The restrained geodetic number of a graph

Abstract A geodetic set S ⊆ V (G) of a graph G = (V, E) is a restrained geodetic set if the subgraph G[V \ S] has no isolated vertex. The minimum cardinality of a restrained geodetic set is the restrained geodetic number. In this paper, we initiate the study of the restrained geodetic number

THE RESTRAINED STEINER NUMBER OF A GRAPH

For a connected graph G = (V, E) of order p, a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets. A set W of vertices of a graph G is a restrained Steiner set if W is a Steiner set, and if either W = V or the subgraph G[V − W ] induced by V − W has no isolated vertices. The minimum cardinality of a restrained Steiner set of G is the restrained Steiner number of G, and is denoted by s r (G). The restrained Steiner number of certain classes of graphs are determined. Connected graphs of order p with restrained Steiner number 2 are characterized. Various necessary conditions for the restrained Steiner number of a graph to be p are given. It is shown that, for integers a, b and p with 4 ≤ a ≤ b ≤ p, there exists a connected graph G of order p such that s(G) = a and s r (G) = b. It is also proved that for every pair of integers a, b with a ≥ 3 and b ≥ 3, there exists a connected graph G with s r (G) = a and g r (G) = b

Potential tsunami hazard associated with the Kerepehi Fault, Hauraki Gulf, New Zealand

A review of 'active' faults in the Auckland region identified the Kerepehi Fault as a potential tsunami source (de Lange and Hull, 1994). The Kerepehi Fault trends NNW, transecting the central region of the inner Hauraki Gulf. A tsunamigenic event along the Kerepehi Fault has not occurred during historical times making this hazard difficult to quantify. Consequently, this study assessed the hazard represented by the Kerepehi Fault using numerical simulation, which required that the locations of offshore segments and behaviour of each segment of the Kerepehi Fault be determined. Shallow seismic sub-bottom profiles totalling 135 km in length were analysed and used to locate the submarine extension of the Kerepehi Fault. Without core data, only the relative timing of movement along the fault could be determined. The seismic data clearly showed that movement has occurred since the last observable reflector/unconformity. Onland evidence indicates that this unconformity corresponds to a surface flooding event which occurred at 6.5 ka; suggesting that the submarine extension of the Kerepehi Fault is still active and potentially tsunamigenic. Seismic data indicates the offshore Kerepehi Fault trends NNW up the central Firth of Thames and is divided into four segments by three WSW-ENE trending transverse faults. The displacement that is most likely to occur for any given event ranges from 2.1-7.35 m. Tsunami generation and propagation was modelled using a linear, finite element model, 'TSUNAMI'; a finite difference hydrodynamic circulation model, '3DD'; and an empirical set of parametric equations (Abe, 1995). The results indicate: (i) The greatest risk to Thames township is associated with displacement along an adjacent fault segment, which produces wave heights of the order of 1.8 m. (ii) The largest shoreline wave heights (~4 m) were generated by displacement in deep water and had the most severe impact upon Pakatoa, Ponui, and Rotorua Islands. The maximum mainland wave resulting from displacement along the Kerepehi Fault impacts at Deadmans Point and has a height of the order of 2.8 m. (iii) The overall tsunami hazard associated with this fault is low. Comparison of wave heights generated by the three methods indicated that for shallow water source regions, 'TSUNAMI' generally under predicts maximum wave height. This is suggested to occur because 'TSUNAMI' is unable to cope with the highly nonlinear wave processes that occur in shallow water. The behaviour of tsunami waves within the Firth of Thames was further investigated through the simulation of teletsunami (distantly generated tsunami) propagation into the Hauraki Gulf, which was undertaken using the model '3DD'. Wave height time histories for selected regions and amplitude attenuation graphs produced from '3DD' output indicate: (i) Generally, for wave periods between 10 and 30 minutes, the maximum predicted sea level rise increases as teletsunami wave period becames longer. (ii) The confined nature of the Firth of Thames acts to focus wave energy resulting in elevated wave heights within this embayment. The maximum amplitude reinforcement occurring in the Firth of Thames is equivalent to 50 % of the amplitude at the open ocean boundary and is considered significant. (iii) For townships adjacent to the Firth of Thames, the maximum rise above mean sea level caused by teletsunamis of similar magnitude to the 1960 Chilean tsunami was between 0.36 and 0.49 m. The largest of these, 0.49 m, is observed at Tapu. (iv) A teletsunami of this magnitude, represents a less significant hazard than tsunamis locally generated along the Kerepehi Fault. Wave heights observed during the 1960 Chilean tsunami and those predicted by model '3DD' are in strong agreement. However, the coarse grid employed by '3DD' (1500 m) limits the number of grid cells per wavelength. Consequently wave propagation in shallow water may not be fully represented, particularly for short period waves. Hence, the above results should be regarded with caution. The probable extent of tsunami inundation occurring in the Thames region was investigated using the non-linear finite difference model, 'TUNAMI N2'. The results indicate that should a tsunami of at least moderate amplitude (3 m) be generated in the Firth of Thames, 7 metres of vertical run-up is likely to occur between Tararu and Moanataiari, and land adjacent to the Thames aerodrome will be horizontally inundated by up to 450 m

VISIR-I: Small vessels - Least-time nautical routes using wave forecasts

A new numerical model for the on-demand computation of optimal ship routes based on sea-state forecasts has been developed. The model, named VISIR (discoVerIng Safe and effIcient Routes) is designed to support decision-makers when planning a marine voyage. The first version of the system, VISIR-I, considers medium and small motor vessels with lengths of up to a few tens of metres and a displacement hull. The model is comprised of three components: a route optimization algorithm, a mechanical model of the ship, and a processor of the environmental fields. The optimization algorithm is based on a graph-search method with time-dependent edge weights. The algorithm is also able to compute a voluntary ship speed reduction. The ship model accounts for calm water and added wave resistance by making use of just the principal particulars of the vessel as input parameters. It also checks the optimal route for parametric roll, pure loss of stability, and surfriding/broaching-to hazard conditions. The processor of the environmental fields employs significant wave height, wave spectrum peak period, and wave direction forecast fields as input. The topological issues of coastal navigation (islands, peninsulas, narrow passages) are addressed. Examples of VISIR-I routes in the Mediterranean Sea are provided. The optimal route may be longer in terms of miles sailed and yet it is faster and safer than the geodetic route between the same departure and arrival locations. Time savings up to 2.7% and route lengthening up to 3.2% are found for the case studies analysed. However, there is no upper bound for the magnitude of the changes of such route metrics, which especially in case of extreme sea states can be much greater. Route diversions result from the safety constraints and the fact that the algorithm takes into account the full temporal evolution and spatial variability of the environmental fields

Modeling and Optimizing for NP-hard Problems in Graph Theory

This PhD thesis introduces optimization methods for graph problems classified as NP-hard. These are problems for which no deterministic algorithm is capable of solving them in polynomial time. More specifically, three graph problems were addressed, and for each, different optimization methods were used. These methods include standard methods, metaheuristics, and heuristics. In all cases, the performance of these methods was compared with those proposed in the literature, considering factors such as execution time and the quality of the solutions achieved. This comparative analysis aims to demonstrate the effectiveness of the proposed optimization methods