Cyclability in Graph Classes

A subset T subseteq V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every subset of vertices of G of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is, a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes

The Parameterized Complexity of Graph Cyclability

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a non-negative integer $k,$ decide whether the cyclability of $G$ is at least $k,$ is {\sf NP}-hard. We study the parametrized complexity of this problem. We prove that this problem, parameterized by $k,$ is {\sf co\mbox{-}W[1]}-hard and that its does not admit a polynomial kernel on planar graphs, unless {\sf NP}\subseteq{\sf co}\mbox{-}{\sf NP}/{\sf poly}. On the positive side, we give an {\sf FPT} algorithm for planar graphs that runs in time $2^{2^{O(k^2\log k)}}\cdot n^2$. Our algorithm is based on a series of graph-theoretical results on cyclic linkages in planar graphs

Cyclability, Connectivity and Circumference

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$ \textit{ordered} vertices are present in a common cycle in that order, then the graph is said to be $k$-ordered. We show that when $k \leq \sqrt{n+3}$, $k$-cyclable graphs also have circumference $c(G) \geq 2k$, and that this is best possible. Furthermore when $k \leq \frac{3n}{4} -1$, $c(G) \geq k+2$, and for $k$-ordered graphs we show $c(G) \geq \min\{n,2k\}$. We also generalize a result by Byer et al. on the maximum number of edges in nonhamiltonian $k$-connected graphs, and show that if $G$ is a $k$-connected graph of order $n \geq 2(k^2+k)$ with $|E(G)| > \binom{n-k}{2} + k^2$, then the graph is hamiltonian, and moreover the extremal graphs are unique

On the k-spanning cyclability of 4-valent Cayley graphs on Abelian groups

A graph $X$ is $k$-spanning cyclable if for any subset $S$ of $k$ distinct vertices there is a 2-factor of $X$ consisting of $k$ cycles such that each vertex in $S$ belongs to a distinct cycle. In this paper we examine the $k$-spanning cyclability of 4-valent Cayley graphs on Abelian groups.Comment: 18 pages, 8 figure

Synthesis of anthraquinone based electroactive polymers: A critical review

Conducting polymers or synthetic monomers have revolutionized the world and are at the heart of scientific research having a scope of vast diverse applications in many technological fields. The conducting and redox polymers have been investigated as energy storage systems because of their better sustainability, ease of synthesis, and environmental compatibility. Owing to the conducting properties of quinones, they gain too much importance among the researchers. Keeping in view the importance and sustainability of conducting polymers, for the first time, this study compiles a detailed overview of synthetic approaches followed by investigations on electrochemical properties and future directions. This study critically examines the synthetic process of simple monomers, substituted monomers, and polymers of anthraquinone (AQ) under the classification of low- and high-molecular-weight AQ–based derivatives, their working principles, and their electrochemical applications, which enable us to explore their novel possible application in automotive, solar cell devices, aircraft aileron, and biomedical equipment. Irrefutably, we confirm that high-molecular-weight polymeric AQ compounds are best in comparison with low-molecular-weight AQ monomers because they have pre-eminent properties over monomeric systems. Because of the significant properties of AQ, polymeric systems are high demanding and have emerged as a hot topic among the researchers these days. In the current scenario, this study is of immense importance because it identifies and discusses the right and sustainable combination and paves the way to utilize these novel materials in different technologies

Connectivity and Cycles in Graphs

https://digitalcommons.memphis.edu/speccoll-faudreerj/1199/thumbnail.jp

Connectivity and Cycles

https://digitalcommons.memphis.edu/speccoll-faudreerj/1191/thumbnail.jp

Cyclability: Combinatorial Properties, Algorithms and Complexity

Ένα γράφημα G καλείται k-κυκλώσιμο, αν για κάθε k από τις κορυφές του υπάρχει ένας κύκλος στο G που τις περιέχει. Η κυκλωσιμότητα ενός γραφήματος G είναι ο μέγιστος ακέραιος k για τον οποίο το G είναι k-κυκλώσιμο και είναι μία παράμετρος που σχετίζεται με τη συνεκτικότητα. Σε αυτή τη διδακτορική διατριβή μελετάμε, κυρίως από τη σκοπιά της Παραμετρικής Πολυπλοκότητας, το πρόβλημα ΚΥΚΛΩΣΙΜΟΤΗΤΑ: Δεδομένου ενός γραφήματος G = (V,E) και ενός μη αρνητικού ακεραίου k (η παράμετρος), να αποφασιστεί αν η κυκλωσιμότητα του G είναι ίση με k. Το πρώτο μας αποτέλεσμα είναι αρνητικό και δείχνει ότι η ύπαρξη ενός FPT-αλγορίθμου για την επίλυση του προβλήματος ΚΥΚΛΩΣΙΜΟΤΗΤΑ είναι απίθανη (εκτός αν FPT = co- W[1], το οποίο θεωρείται απίθανο). Πιο συγκεκριμένα, αποδεικνύουμε ότι το πρόβλημα ΚΥΚΛΩΣΙΜΟΤΗΤΑ είναι co-W[1]-δύσκολο, ακόμα και αν περιορίσουμε την είσοδο στο να είναι χωριζόμενο γράφημα. Από την άλλη, δίνουμε έναν FPT-αλγόριθμο για το ίδιο πρόβλημα περιορισμένο στην κλάση των επίπεδων γραφημάτων. Για να το πετύχουμε αυτό αποδεικνύουμε μια σειρά από συνδυαστικά αποτελέσματα σχετικά με την κυκλωσιμότητα και εφαρμόζουμε μια εκδοχή δύο βημάτων της περίφημης τεχνικής της άσχετης κορυφής, που εισήχθη από τους Robertson και Seymour στη σειρά εργασιών τους για Ελλάσονα Γραφήματα, ως ένα κρίσιμο συστατικό του αλγορίθμου τους για την επίλυση του προβλήματος των ΔΙΑΚΕΚΡΙΜΕΝΩΝ ΜΟΝΟΠΑΤΙΩΝ. Για να αποδείξουμε την ορθότητα του αλγορίθμου μας εισάγουμε έννοιες, όπως αυτή των ζωτικών κυκλικών συνδέσμων, και αποδεικνύουμε αποτελέσματα με ανεξάρτητου γραφοθεωρητικού ενδιαφέροντος. Κλείνουμε τη μελέτη μας με ένα δεύτερο αρνητικό αποτέλεσμα: Αποδεικνύουμε ότι για το πρόβλημα της ΚΥΚΛΩΣΙΜΟΤΗΤΑΣ δεν υπάρχουν πολυωνυμικοί πυρήνες, ακόμα και αν περιοριστούμε σε κυβικά επίπεδα γραφήματα, εκτός και αν δεν ισχύει μια υπόθεση της κλασσικής Θεωρίας Πολυπλοκότητας (ότι NP υποσύνολο του co-NP/poly).A graph G is called k-cyclable, if for every k of its vertices there exists a cycle in G that contains them. The cyclability of G is the maximum integer k for which G is k-cyclable and it is a connectivity related graph parameter. In this doctoral thesis we study, mainly from the Parameterized Complexity point of view, the Cyclability problem: Given a graph G = (V,E) and an integer k (the parameter), decide whether the cyclability of G is equal to k. Our first result is a negative one and shows that the existence of an FPT-algorithm for solving Cyclability is unlikely (unless FPT = co-W[1], which is considered unlikely). More specifically, we prove that Cyclability is co-W[1]-hard, even if we restrict the input to be a split graph. On the other hand, we give an FPT-algorithm for the same problem when restricted to the class of planar graphs. To do this, we prove a series of combinatorial results regarding cyclability and apply a two-step version of the so called irrelevant vertex technique, which was introduced by Robertson and Seymour in their Graph Minors series (Irrelevant vertices in linkage problems) as a crucial ingredient for their algorithm solving the Disjoint Paths problem. To prove the correctness of our algorithm, we introduce notions, like the one of vital cyclic linkages, and give results of independent graph-theoretic interest. We conclude our study with a negative result: We prove that Cyclability admits no polynomial kernel, even when restricted to cubic planar graphs, unless a classical complexity theoretic assumption (that NP is a subset of co-NP/poly) fails

A sufficient condition for pancyclability of graphs

AbstractLet G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by dS(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with dS(u,v)=2, max{d(u),d(v)}≥n2, then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with dS(u,v)=2, max{d(u),d(v)}≥n+12, then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221–227] for the case when S=V(G)