Showing papers in "Discrete Mathematics, Algorithms and Applications in 2012"
TL;DR: The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks.
Abstract: Circulant graphs have been extensively investigated over the past 30 years because of their broad application to different fields of theory and practice. Two known surveys on circulant networks including a survey on undirected circulants have been published: by Bermond et al. [Distributed loop computer networks: A survey, J. Parallel Distributed Comput.24 (1995) 2–10] and by Hwang [A survey on multi-loop networks, Theoret. Comput. Sci.299 (2003) 107–121]. The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks. We focus on the survey connected with study of structural and communicative properties of circulant networks.
57 citations
TL;DR: It is shown that if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z (H)|, thenH and G have isomorphic conjugate graphs.
Abstract: In this paper we introduce the conjugate graph $\Gamma^{c}_{G}$ associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if $\Gamma^{c}_{G}\cong \Gamma^{c}_{S}$, where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs.
33 citations
TL;DR: It is obtained that (1) ndiΣ(G) ≤ max{2Δ (G) + 1, 25} if G is a planar graph, (2) nDi(G), 19 if G was a graph such that mad(G%) ≤ 5.
Abstract: A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max{2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max{2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.
29 citations
TL;DR: A vector space partition is a collection of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of.
Abstract: A vector space partition is here a collection of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of . Vector space partitions relate to finite projective planes, design theory and error correcting codes. In the first part of the paper I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the paper contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V(n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition. Finally, I will also give a few historical remarks.
25 citations
TL;DR: It is proved that, corresponding to each edge of the Theta graph Θ6, there is a path in the Yao graph Y6 whose length is at most 8.82 times the edge length, obtaining an upper bound of 17.64 on the stretch factor of Y6.
Abstract: Yao and Theta graphs are defined for a given point set and a fixed integer k > 0. The space around each point is divided into k cones of equal angle, and each point is connected to a nearest neighbor in each cone. The difference between Yao and Theta graphs is in the way the nearest neighbor is defined: Yao graphs minimize the Euclidean distance between a point and its neighbor, and Theta graphs minimize the Euclidean distance between a point and the orthogonal projection of its neighbor on the bisector of the hosting cone. We prove that, corresponding to each edge of the Theta graph Θ6, there is a path in the Yao graph Y6 whose length is at most 8.82 times the edge length. Combined with the result of Bonichon et al., who prove an upper bound of 2 on the stretch factor of Θ6, we obtain an upper bound of 17.64 on the stretch factor of Y6.
22 citations
TL;DR: A new proof of the same theorem, which works for all graphs, is given, which settles completely the second part of Theorem 2.10(b) in that paper for all values a, b, c with 1 ≤ a ≤ b ≤ c, which also solves the Problem 2.12.
Abstract: In the paper entitled, "Monophonic Distance in Graphs", which appeared in Discrete Mathematics, Algorithms and Applications3(2) (2011) 159–169, it was proved (Theorem 3.4) that every graph is the monophonic center of some connected graph. Although the statement of the theorem is true, the proof given works only for graphs with monophonic diameter at most 2. Hence, we give a new proof of the same theorem, which works for all graphs. We settle completely the second part of Theorem 2.10(b) in that paper for all values a, b, c with 1 ≤ a ≤ b ≤ c, which also solves the Problem 2.12 given in that paper.
21 citations
TL;DR: It is shown that for any e > 0, a (1 + e)-approximation can be found in O((1/e)n3) time, i.e., it admits a fully polynomial-time approximation scheme (FPTAS).
Abstract: We study how to partition an interval graph with non-negative vertex weights into k connected subgraphs such that the minimum total weight of any part of the partition is maximized. For k = 2, it is shown that for any e > 0, a (1 + e)-approximation can be found in O((1/e)n3) time, i.e., it admits a fully polynomial-time approximation scheme (FPTAS). For any fixed k > 2, the problem also admits an FPTAS when restricted to k-connected interval graphs.
19 citations
TL;DR: A unified approach is adopted to solve the minimum signed (minus) total domination problem for chordal bipartite graphs in O(n + m) time and the method is also able to solved the minimum k-tuple total domination problems for chordals bipartites graphs in N + m time.
Abstract: In this paper, we consider minimum total domination problem along with two of its variations namely, minimum signed total domination problem and minimum minus total domination problem for chordal bipartite graphs. In the minimum total domination problem, the objective is to find a smallest size subset TD ⊆ V of a given graph G = (V, E) such that |TD∩NG(v)| ≥ 1 for every v ∈ V. In the minimum signed (minus) total domination problem for a graph G = (V, E), it is required to find a function f : V → {-1, 1} ({-1, 0, 1}) such that f(NG(v)) = ∑u∈NG(v)f(u) ≥ 1 for each v ∈ V, and the cost f(V) = ∑v∈V f(v) is minimized. We first show that for a given chordal bipartite graph G = (V, E) with a weak elimination ordering, a minimum total dominating set can be computed in O(n + m) time, where n = |V| and m = |E|. This improves the complexity of the minimum total domination problem for chordal bipartite graphs from O(n2) time to O(n + m) time. We then adopt a unified approach to solve the minimum signed (minus) total domination problem for chordal bipartite graphs in O(n + m) time. The method is also able to solve the minimum k-tuple total domination problem for chordal bipartite graphs in O(n + m) time. For a fixed integer k ≥ 1 and a graph G = (V, E), the minimum k-tuple total domination problem is to find a smallest subset TDk ⊆ V such that |TDk ∩ NG(v)| ≥ k for every v ∈ V.
18 citations
TL;DR: It is proved that given a set of n sensors, each equipped with directional antennae with any angle of transmission, these antennae can be oriented in such a way that the resulting communication structure is a strongly connected digraph spanning all n sensors.
Abstract: Traditional approaches to connectivity in sensor networks are based on the omnidirectional antenna model which relies on the assumption that the sensors send and receive in all directions. Current technologies make possible the utilization of sensors with directional antenna capabilities whereby the sensors send and/or receive along a sector of a predefined angle (or beam-width). Although several researchers in the scientific literature have investigated the impact of directional antennae on network throughput, energy consumption, as well as security very little is known concerning the effect of directional antennae on its connectivity. In this paper, we introduce for the first time a new sensor model with each sensor being able to transmit in any one of k directions, for some fixed k, and explore the algorithmic limits and potential of such a directional antenna model. More specifically, given a set of n sensors in the plane, we consider the problem of establishing a strongly connected ad hoc network from these sensors using directional antennae. In particular, we prove that given such set of sensors, each equipped with k, 1 ≤ k ≤ 5, directional antennae with any angle of transmission, these antennae can be oriented in such a way that the resulting communication structure is a strongly connected digraph spanning all n sensors. Moreover, the transmission range of the antennae is at most times the optimal range (a range necessary to establish a connected network on the same set of sensors using omnidirectional antennae). The algorithm which constructs this orientation runs in O(n) time provided a minimum spanning tree on the set of sensors is given. We show that our solution can be used to give a tradeoff on the range and angle when each sensor has one antenna. Further, we also prove that for two antennae it is NP-hard to decide whether such an orientation exists if both the transmission angle and range are small for each antennae.
18 citations
TL;DR: A novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation is introduced, and by using the class of P4-sparse graphs as the relaxed graph class, efficient bounded search tree algorithms are obtained for several parametrized deletion problems.
Abstract: Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of P4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parametrized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [R. Niedermeier and P. Rossmanith, An efficient fixed-parameter algorithm for 3-hitting set, J. Discrete Algorithms1(1) (...
17 citations
TL;DR: A dynamic programming-based O(n + m) time algorithm to find a maximum cardinality acyclic matching in a chain graph having n vertices and m edges and obtain an expression for the number of maximum Cardinality acYclic matchings in achain graph is proposed.
Abstract: A set M ⊆ E is called an acyclic matching of a graph G = (V, E) if no two edges in M are adjacent and the subgraph induced by the set of end vertices of the edges of M is acyclic. Given a positive integer k and a graph G = (V, E), the acyclic matching problem is to decide whether G has an acyclic matching of cardinality at least k. Goddard et al. (Discrete Math.293(1–3) (2005) 129–138) introduced the concept of the acyclic matching problem and proved that the acyclic matching problem is NP-complete for general graphs. In this paper, we propose an O(n + m) time algorithm to find a maximum cardinality acyclic matching in a chain graph having n vertices and m edges and obtain an expression for the number of maximum cardinality acyclic matchings in a chain graph. We also propose a dynamic programming-based O(n + m) time algorithm to find a maximum cardinality acyclic matching in a bipartite permutation graph having n vertices and m edges. Finally, we strengthen the complexity result of the acyclic matching problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs.
TL;DR: A tabu search algorithm (TSA) is presented for the construction of mixed covering arrays (MCAs), and the fine tuning process used to identify the best parameter values for TSA is reported.
Abstract: The development of a new software system involves extensive tests of the software functionality in order to identify possible failures. Also, a software system already built requires a fine tuning of its configurable options to give the best performance in the environment where it is going to work. Both cases require a finite set of tests that avoids testing all the possible combinations (which is time consuming); to this situation mixed covering arrays (MCAs) are a feasible alternative. MCAs are combinatorial structures having a case per row. MCAs are small, in comparison with exhaustive search, and guarantee a level of interaction among the involved parameters (a difference with random testing). We present a tabu search algorithm (TSA) for the construction of MCAs. Also, we report the fine tuning process used to identify the best parameter values for TSA. The analyzed TSA parameters were three different initialization functions, five different tabu list sizes and the mixture of four neighborhood functio...
TL;DR: A new proof is presented for a well-known inequality that gives an upper bound for the number of nonoverlapping unit discs whose centers can be packed into a compact convex region.
Abstract: In this paper, we present a new proof for a well-known inequality, conjectured by Zassenhaus in 1947 and proved independently by Groemer in 1960 and Oler in 1961 The inequality gives an upper bound for the number of nonoverlapping unit discs whose centers can be packed into a compact convex region, and recently obtains a lot of applications in study of sensor networks
TL;DR: This paper characterize n-vertex unicyclic graphs with girth k, having minimal Gutman index.
Abstract: Let G be a connected graph with vertex set V(G). The Gutman index of G is defined as S(G) = ∑{u, v}⊆V(G) d(u)d(v)d(u, v), where d(u) is the degree of vertex u, and d(u, v) denotes the distance between u and v. In this paper, we characterize n-vertex unicyclic graphs with girth k, having minimal Gutman index.
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TL;DR: This paper is solving recurrences of the expected number of key comparisons and exchanges performed by the dual pivot Quicksort, obtaining the exact and asymptotic total average values contributing to its time complexity.
Abstract: In this paper, we analyze the dual pivot Quicksort, a variant of the standard Quicksort algorithm, in which two pivots are used for the partitioning of the array. We are solving recurrences of the expected number of key comparisons and exchanges performed by the algorithm, obtaining the exact and asymptotic total average values contributing to its time complexity. Further, we compute the average number of partitioning stages and the variance of the number of key comparisons. In terms of mean values, dual pivot Quicksort does not appear to be faster than ordinary algorithm.
TL;DR: It is proved that for any graph G, , where deg(G) denotes the degeneracy of G, that is, the minimum number k such that δ(H) ≤ k for any subgraph H of G that ρl (G) ≤ 3 for any planar graph G.
Abstract: Let G be a graph. The point arboricity of G, denoted by ρ(G), is the minimum number of colors that can be used to color the vertices of G so that each color class induces an acyclic subgraph of G. Borodin et al. (Discrete Math.214 (2000) 101–112) first introduced the list point arboricity of G, denoted by ρl(G). We prove that for any graph G, , where deg(G) denotes the degeneracy of G, that is, the minimum number k such that δ(H) ≤ k for any subgraph H of G. Using this upper bound, we show that ρl(G) ≤ 3 for any planar graph G. In particular, if either G is K4-minor free, or for an integer k ∈ {3, 4, 5, 6}, G is planar and does not contain k-cycles, then ρl(G) ≤ 2. For any graph G of order n, . In addition, we provide a new proof of a theorem of Borodin et al., which states that if G is neither a complete graph of odd order nor a cycle then . Finally, we show that la(G) = lla(G) = 2 if G is 3-regular, and la(G) = lla(G) = 3 if G is 4-regular, where la(G) is the linear arboricity of G and lla(G) is list linear arboricity of G which is introduced recently by An and Wu.
TL;DR: It is proved that the Erdos–Faber–Lovasz conjecture is true for all linear hypergraphs on n vertices with .
Abstract: The celebrated Erdos–Faber–Lovasz conjecture originated in the year 1972. It can be stated as follows: any linear hypergraph on n vertices has chromatic index at most n. Different formulations of the conjecture have been obtained and the conjecture is proved to be true in some particular cases. But the problem is still unsolved in general. In this paper, we prove that the conjecture is true for all linear hypergraphs on n vertices with . It generalizes an existing result regarding an equivalent formulation of the conjecture for dense hypergraphs [A. Sanchez-Arroyo, The Erdos–Faber–Lovasz conjecture for dense hypergraphs, Discrete Math.308 (2008) 991–992].
TL;DR: In the Euclidean TSP with neighborhoods (TSPN) with neighborhoods, this paper presented several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines.
Abstract: In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length L.
TL;DR: It is shown that if G is a planar graph with g(G) ≥ 7 and Δ(G), then χi (G) ≤ Δ( G) + 1, which is the injective chromatic number of G.
Abstract: An injective k-coloring of a graph G is an assignment of k colors to the vertices of G so that any two vertices with a common neighbor receive distinct colors, and χi(G) is the injective chromatic number of G. Dimitrov et al. proved χi(G) ≤ Δ(G) + 2 for a planar graph G with g(G) ≥ 7. In this paper, we show that if G is a planar graph with g(G) ≥ 7 and Δ(G) ≥ 7, then χi(G) ≤ Δ(G) + 1.
TL;DR: This paper presents two dynamic programming algorithms and then designs an FPTAS for the considered problem, scheduling deteriorating jobs on a single machine with release times and rejection.
Abstract: This paper considers scheduling deteriorating jobs on a single machine with release times and rejection. Deteriorating job means that its actual processing time is a increasing function on its execution starting time. In this situation, jobs can be rejected by paying penalties. Each job is associated with a release time. The objective is to minimize the makespan plus the total penalty incurred by rejecting jobs. We present two dynamic programming algorithms and then design an FPTAS for the considered problem.
TL;DR: In this paper, the antimedian function is characterized for two classes of graphs on which the Antimedian is well behaved: paths and hypercubes, where the median function does not have a nice behavior on most classes.
Abstract: An antimedian of a profile π = (x1, x2, …, xk) of vertices of a graph G is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on G and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The 'converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for two classes of graphs on which the antimedian is well behaved: paths and hypercubes.
TL;DR: The distance spectral radius of bicyclic graphs in $\mathcal{B}_{n}$ is studied, and the graph with the largestdistance spectral radius is determined.
Abstract: Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let Pp+1 = x1x2⋯xp+1, Pt+1 = y1y2⋯yt+1 and Pq+1 = z1z2⋯zq+1 be three vertex-disjoint paths. Identifying the initial vertices as u0 and the terminal vertices as v0, the resultant graph, denoted by θ(p; t; q), is called a θ-graph. Let $\mathcal{B}_{n}$ be the class of all bicyclic graphs on n vertices, which contain a θ-graph as an induced subgraph. In this paper, we study the distance spectral radius of bicyclic graphs in $\mathcal{B}_{n}$, and determine the graph with the largest distance spectral radius.
TL;DR: This paper addresses the problem of augmenting the edge set of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected and proves that knowledge of vertex coordinates is crucial to this construction.
Abstract: 2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any .
TL;DR: A novel iterative algorithm of calculating the exact transitive closure of a parametrized graph being represented by a union of simple affine integer tuple relations is presented and the effectiveness of the presented algorithm is compared with those of related ones.
Abstract: A novel iterative algorithm of calculating the exact transitive closure of a parametrized graph being represented by a union of simple affine integer tuple relations is presented. When it is not possible to calculate exact transitive closure, the algorithm produces its upper bound. To calculate the transitive closure of the union of all simple relations, the algorithm recognizes the class of each simple relations, calculates its exact transitive closure, forms the union of calculated transitive closures, and applies this union in an iterative procedure. Results of experiments aimed at the comparison of the effectiveness of the presented algorithm with those of related ones are outlined and discussed.
TL;DR: This paper constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes, and other monoid codes through monoid ring B such that b = a + 1, where a is any positive integer.
Abstract: Let B[X; S] be a monoid ring with any fixed finite unitary commutative ring B and $\frac{a}{b}{\mathbb Z}_{0}$ is the monoid S such that b = a + 1, where a is any positive integer. In this paper we constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes through monoid ring $B[X;\frac{a}{b}{\mathbb Z}_{0}]$. For a = 1, almost all the results contained in [16] stands as a very particular case of this study.
TL;DR: The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than γ(G), where γ (G) denote the domination number of G.
Abstract: The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than γ(G), where γ(G) denote the domination number of G. Let G be a toroidal graph with maximum degree Δ(G). In this paper, we show that b(G) ≤ 9. Moveover, if Δ(G) ≠ 6, then b(G) ≤ 8.
TL;DR: The exact values for the signed star domination number for certain classes of Cayley digraphs and Cayley graphs are obtained.
Abstract: Let G be a simple connected graph with vertex set V(G) and edge set E(G). A function f : E(G) → {-1, 1} is called a signed star dominating function (SSDF) on G if ∑e∈E(v) f(e) ≥ 1 for every v ∈ V(G), where E(v) is the set of all edges incident to v. The signed star domination number of G is defined as γSS(G) = min{∑e∈E(G) f(e) | f is a SSDF on G}. In this paper, we obtain exact values for the signed star domination number for certain classes of Cayley digraphs and Cayley graphs.
TL;DR: This paper has the following contributions: if the highest value h among all vi(x) is known in advance, it is shown the lower bound of the competitive ratio is ⌊log h⌋/2, then an online algorithm with competitive ratio 4⌊ log h⋅ + 6 is given.
Abstract: Given a seller with m items, a sequence of users {u1, u2, …} come one by one, the seller must set the unit price and assign some items to each user on his/her arrival. Items can be sold fractionally. Each ui has his/her value function vi(⋅) such that vi(x) is the highest unit price ui is willing to pay for x items. The objective is to maximize the revenue by setting the price and number of items for each user. In this paper, we have the following contributions: if the highest value h among all vi(x) is known in advance, we first show the lower bound of the competitive ratio is ⌊log h⌋/2, then give an online algorithm with competitive ratio 4⌊log h⌋ + 6; if h is not known in advance, we give an online algorithm with competitive ratio 2⋅hlog-1/2 h + 8⋅h3log-1/2 h.
TL;DR: In this article, it was shown that every g-convex set of a connected graph can be recognized in polynomial time, and a new convexity-theoretic characterization of Ptolemaic graphs can be found.
Abstract: Let G be a connected graph. A subset X of V(G) is g-convex (m-convex) if it contains all vertices on shortest (induced) paths between vertices in X. We state characteristic properties of graphs in which every g-convex set is m-convex, based on which we show that such graphs can be recognized in polynomial time. Moreover, we state a new convexity-theoretic characterization of Ptolemaic graphs.
TL;DR: In this article, a necessary and sufficient condition for existence of a contractible Hamiltonian cycle in the edge graph of equivelar maps on surfaces is presented. And an algorithm is presented to find such cycles (if they exist).
Abstract: We present a necessary and sufficient condition for existence of a contractible Hamiltonian cycle in the edge graph of equivelar maps on surfaces. We also present an algorithm to find such cycles (if they exist). This is further generalized and shown to hold for more general maps.