Cycles In Graphs Research Articles

Double domination number of the shadow (2,3)-distance graphs

Let G (V,E) be a graph with the vertex set V (G) and S be a subset of V(G). If every vertex of V is dominated by S at least twice, then the set S is called a double domination set of the graph. The number of elements of the double domination set with the smallest cardinality is called double domination number and denoted by 2 (G)  notation. In this paper, we discussed the double domination parameter on some types of shadow distance graphs such as cycle, path, star, complete bipartite and wheel graphs.

A solution to Erdős and Hajnal’s odd cycle problem

In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let C ( G ) \mathcal {C}(G) be the set of cycle lengths in a graph G G and let C o d d ( G ) \mathcal {C}_{\mathrm {odd}}(G) be the set of odd numbers in C ( G ) \mathcal {C}(G) . We prove that, if G G has chromatic number k k , then ∑ ℓ ∈ C o d d ( G ) 1 / ℓ ≥ ( 1 / 2 − o k ( 1 ) ) log ⁡ k \sum _{\ell \in \mathcal {C}_{\mathrm {odd}}(G)}1/\ell \geq (1/2-o_k(1))\log k . This solves Erdős and Hajnal’s odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some d d such that each graph with chromatic number at least d d (or perhaps even only average degree at least d d ) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every k k , there is some d d so that every graph with average degree at least d d has a subdivision of the complete graph K k K_k in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.

Technical safety of buildings around polish quarries against harmful seismic vibrations caused by rock blasting

This paper presents the results of measurements of the elliptical distribution of seismic vibrations generated during rock blasting.The technical safety of a building in the area of seismic vibrations is determined by the vibration velocity that does not exceed the line B included in zone II of the scale of dynamic influences [SWD]. The hypothesis proved in the paper is that for both elliptical and circular distributions of the vibration velocity in rock mining with blasting material [BM] there is a directionality of the horizontal radial component Vx and the tangential component Vy of the vibration velocity.The magnitude of the components Vx and Vy of the vibration velocity at the same distance from the vibration source depends on the directional angle "α" between the line of the blast holes and the line connecting the centre of the surface of the rock block being mined with [BM] and the place of measurement. The elliptical shapes of the graphs Vx and Vy obtained during the mining of the BM of the basalt deposit, which depend on the value of the directional angle, were compared with the shape of the circular velocity graphs Vx and Vy. New correlations are given that take into account the directional angle when calculating the maximum vibration velocity values needed to determine the damage caused to a bulding by the SWD.A dynamic impact scale [SWD] for the assessment of building damage as the vibration velocity acting on the building increases is given and discussed. Vibration velocity diagrams measured during the excavation of BM rock are presented for circular distributions in accordance with theoretical predictions. A vibration velocity diagram for an elliptical distribution, inconsistent with a circular distribution, measured during the excavation of BM rock is presented. It is shown that the directionality of the horizontal radial and tangential components of vibration velocity exists for both circular and elliptical distribution of vibration velocity in rock excavation with BM. In the conclusions, for the elliptical velocity field of the vibrations, the conditions to be observed in order to be able to direct the smallest vibrations towards the built up area are given.

Bipartite Domination Number of Mycielski Graph of Some Graph Families

For a nontrivial connected graph G, a non-empty set S \(\subseteq\) V (G) is a bipartite dominating set of graph G, if the subgraph G[S] induced by S is bipartite and for every vertex not in S is dominated by any vertex in S. The bipartite domination number denoted by \(\gamma\)bip(G) of graph G is the minimum cardinality of a bipartite dominating set G. In this paper, we determine the exact bipartite domination number of a crown graph and its mycielski graph as well as the bipartite domination number of the mycielski graph of path and cycle graphs.

Distance Version of Forgotten Topological Index of Some Graph Products

In this paper, we have been found the exact values of distance version of Forgotten Topological index of various types of products such as strong, corona and tensor products of simple and connected graphs. Also we have been calculate the exact values of distance version of Forgotten Topological index of tensor product of path, cycle and complete graphs.

On the number of edges of cyclic subgroup graphs of finite groups

In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds for arbitrary finite groups.

On secure domination number of generalized Mycielskian of some graphs

Generalized Mycielskians are triangle-free networks with a large chromatic number, having a number of desirable characteristics like fast multi-path communication, high fault tolerance, reliable resource sharing, etc. Graph invariants like domination number and secure domination number can be used to protect the network by monitoring each node or moving to a failing node. A dominating set of a graph G is a subset of its vertex set, which can monitor every other vertex of the graph, and γ (G), the domination number of G, is the least cardinality among all dominating sets of G. A secure dominating set S of G is a dominating set with an additional property that, for each vertex u not in S there is a vertex v in S adjacent to u such that, the swapped set (S - {v}) ∪ {u} is dominating. γs (G) is the secure domination number of G, which is the minimum cardinality among all secure dominating sets of G. In this paper, we analyzed γ and γs of the generalized Mycielskian (μm) for path graphs Pn and cycle graphs Cn, varying both n and m, and used these results to obtain bounds for general graphs G.

Coexistence times in the Moran process with environmental heterogeneity

Populations evolve in spatially heterogeneous environments. While a certain trait might bring a fitness advantage in some patch of the environment, a different trait might be advantageous in another patch. Here, we study the Moran birth–death process with two types of individuals in a population stretched across two patches of size N , each patch favouring one of the two types. We show that the long-term fate of such populations crucially depends on the migration rate μ between the patches. To classify the possible fates, we use the distinction between polynomial (short) and exponential (long) timescales. We show that when μ is high then one of the two types fixates on the whole population after a number of steps that is only polynomial in N . By contrast, when μ is low then each type holds majority in the patch where it is favoured for a number of steps that is at least exponential in N . Moreover, we precisely identify the threshold migration rate μ ⋆ that separates those two scenarios, thereby exactly delineating the situations that support long-term coexistence of the two types. We also discuss the case of various cycle graphs and we present computer simulations that perfectly match our analytical results.

Computation of the Distance-Based Bound on Strong Structural Controllability in Networks

In this article, we study the problem of computing a tight lower bound on the dimension of the strong structurally controllable subspace (SSCS) in networks with Laplacian dynamics. The bound is based on a sequence of vectors containing the distances between leaders (nodes with external inputs) and followers (remaining nodes) in the underlying network graph. Such vectors are referred to as the distance-to-leaders vectors. We give exact and approximate algorithms to compute the longest sequences of distance-to-leaders vectors, which directly provide distance-based bounds on the dimension of SSCS. The distance-based bound is known to outperform the other known bounds (for instance, based on zero-forcing sets), especially when the network is partially strong structurally controllable. Using these results, we discuss an application of the distance-based bound in solving the leader selection problem for strong structural controllability. Further, we characterize strong structural controllability in path and cycle graphs with a given set of leader nodes using sequences of distance-to-leaders vectors. Finally, we numerically evaluate our results on various graphs.

Abstract OT2-09-01: TBCRC-055: A Phase II Study of NirAparib, Dostarlimab, and Radiotherapy in Metastatic, PD-L1 Negative or Immunotherapy-Refractory Triple-Negative Breast Cancer (NADiR) – NCT04837209

Abstract Background: Metastatic triple negative breast cancer (mTNBC) is associated with high recurrence and mortality rates. Prior studies have shown an immune checkpoint inhibitor (ICI) + chemotherapy improves progression-free survival for patients with PD-L1 positive mTNBC. There remains a need for treatment options in patients who do not respond to ICI or are PD-L1 negative. Preclinical data suggests that a PARP inhibitor (PARPi) may promote innate immune signaling, and combination with an ICI has shown a positive response in patients with mTNBC. Radiotherapy (RT) is a potent immune stimulator and used for local control in the setting of metastatic breast cancer. This phase II study (NCT04837209) proposes combination of PARPi, ICI, and RT to combat ICI resistance and improve response rates in patients with mTNBC who are PD-L1 negative or who have progressed on prior ICI. Methods: 32 patients with mTNBC defined as ER< 1%, PR< 1%, HER-2-neu 0-1+ by IHC or non-FISH-amplified or patients with metastatic HR+/HER2- breast cancer are anticipated to participate. Eligibility criteria for mTNBC patients includes those who are PD-L1 negative or have progressed on prior ICI. Eligibility criteria for HR+/HER2- patients is specific to those who harbor a deleterious BRCA1 or BRCA2 mutation with or without high tumor mutational burden (TMB). All trial patients should have at least 1 lesion amenable to RT and at least 1 measurable lesion that will not be radiated. Study treatment consists of 3-week cycles, with 500mg dostarlimab given on day 1 of each cycle through cycle 5, then 1000mg given every 6 weeks. RT (24 Gy) is delivered in 3 consecutive fractions starting day 1 of cycle 1. Niraparib (200mg) is dosed orally daily. Tumor biopsies are taken within 28 days pre-treatment, and at C3D1-8. Blood samples are taken at baseline and every odd cycle for cfDNA and PBMC analysis. The primary endpoint is to assess overall response rate as measured by RECIST v1.1 of the combination of niraparib, dostarlimab, and RT. Secondary objectives include assessing safety and toxicity, overall survival, progression free survival, and quality of life. Results: To date, this study has accrued 4 subjects, including 3 with mTNBC, and 1 with HR+/HER2-/BRCA mutant + TMB high mBC. The study is currently open at MGH and Sibley, and the addition of UPenn, Johns Hopkins, and Duke are in progress. Funding for this study was provided by GSK. GSK was provided the opportunity to review a preliminary version of this abstract for factual accuracy, but the authors are solely responsible for final content and interpretation. People with specific interest in the trial should reach out to Elizabeth Scott, Clinical Research Coordinator, at ecscott@mgh.harvard.edu. Citation Format: Elizabeth Scott, Steven J. Isakoff, Alphonse G. Taghian, Jean Wright, Cesar Augusto Santa-Maria, Payal Shah, Neil Taunk, Carey Anders, Rachel Blitzblau, Gaorav Gupta, Alice Ho. TBCRC-055: A Phase II Study of NirAparib, Dostarlimab, and Radiotherapy in Metastatic, PD-L1 Negative or Immunotherapy-Refractory Triple-Negative Breast Cancer (NADiR) – NCT04837209 [abstract]. In: Proceedings of the 2022 San Antonio Breast Cancer Symposium; 2022 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2023;83(5 Suppl):Abstract nr OT2-09-01.

Complexity of Ck-coloring in hereditary classes of graphs

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G)→V(H) such that for every edge uv∈E(G) it holds that f(u)f(v)∈E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of Pt-free graphs.We show that for every odd k≥5, the Ck-Coloring problem, even in the list variant, can be solved in polynomial time in P9-free graphs. The algorithm extends to the list version of Ck-Coloring, where k≥10 is an even number.On the other hand, we prove that if some component of F is not a subgraph of a subdivided claw, then the following problems are NP-complete in F-free graphs:1.the precoloring extension version of Ck-Coloring for every odd k≥5;2.the list version of Ck-Coloring for every even k≥6.

Bipartite Domination in Some Classes of Graphs

For a nontrivial connected graph G, a non-empty set S \(\subseteq\) V (G) is a bipartite dominating set of graph G, if the subgraph G[S] induced by S is bipartite and for every vertex not in S is dominated by any vertex in S. The bipartite domination number denoted by \(\gamma\)bip(G) of graph G is the minimum cardinality of a bipartite dominating set G. In this paper, we determine the exact bipartite domination number of path graph and cycle graph via congruence modulo. Moreover, this study generates the possible exact values of the bipartite domination number of the complete graph, complete bipartite graph, join graph, fan graph and wheel graph.

Progressive Content Generation Based on Cyclic Graph for Generate Dungeon

Dungeon is level in game consisting collection of rooms and doors with obstacles inside. To make good level, takes a lot of time. With Procedural Content Generation (PCG), dungeons can be created automatically. One of the approaches in PCG to create levels is progressive. Progressive approach produces timeline as representation of the interactions in the game. Timeline representation that is in the form of one straight line is good for endless runner, but for dungeon, the levels are linear. In this research, the timeline is changed to cyclic graph. Cyclic graph is formed using graph grammar algorithm. This research aims to build dungeon that has not linear and minimal dead ends. To eliminate linearity in dungeons, branching in dungeons needs to be formed. The steps carried out in this research are designing graph grammar rules, generating population of graphs, evaluating graphs with fitness values, and building dungeons. Four functions are used to determine the fitness value: shortest vertices, average duration, replayability, and variation. Dungeons produced with progressive approach manage to minimize linearity in dungeons. Dungeon formation is very dependent on the rule grammar that forms it. With the evaluation process, linear dungeons resulting from grammar rules can be minimized.

On Molecular Structural Characterization of Cyclen Cored Dendrimers

Macromolecules are gaining much attention in various fields today. Dendrimers are artificially synthesized macromolecules by convergent or divergent approach. They are compact regular structures with spherical dimension and has a vast number of application in disparate fields such as drug delivery, material science, and biology, magnetic resonance imaging, an organic light-emitting device, etc. Determining the pharmacological, chemical, and biological characteristics of a substance necessitates a significant amount of effort. From the chemical graph of the dendrimer structure, we can infer those characteristics with the help of numerical descriptors known as the topological index. The Wiener and Szeged indices are two important distance-based topological indices applicable in nanoscience. The degree-based topological indices also have great importance and huge applications in structural chemistry. These indices together with graph entropy are found to be more effective and have found application in different sciences. In this work, the Wiener index, Szeged indices, Mostar indices, and Padmakar Ivan index for cyclen cored dendrimers are evaluated by converting the original graph into quotient graphs using Θ * - classes. This technique is applied since the regular cut method is a lengthy process while moving on to higher generations and also due to the presence of odd cycles in the structure. The degree-based indices and the degree-based graph entropies for the cyclen cored dendrimers are further studied. The comparison graphs with respect to the topological indices as well as graph entropies are also presented.

On the regularity of small symbolic powers of edge ideals of graphs

Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $s\geq 1$, $$ \mathrm{reg} (I(G)^{(s+1)})\leq \max \bigl \{\mathrm{reg} (I(G))$$ $$+2s, \mathrm{reg} \bigl (I(G)^{(s+1)}+I(G)^s\bigr )\bigr \}. $$ As a consequence, we conclude that $\mathrm{reg} (I(G)^{(2)})\leq \mathrm{reg} (I(G))+2$, and $\mathrm{reg} (I(G)^{(3)})\leq \mathrm{reg} (I(G))+4$. Moreover, it is shown that if for some integer $k\geq 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then $\mathrm{reg} (I(G)^{(s)})\leq 2s+\mathrm{reg} (I(G))-2$, for every integer $s\leq k+1$. Finally, it is proven that $\mathrm{reg} (I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline {G}$ is chordal.

On total coloring the direct product of cycles and bipartite direct product of graphs

A k-total coloring of a graph G is an assignment of k colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which the graph G has a k-total coloring. Clearly, this number is at least Δ(G)+1, where Δ(G) is the maximum degree of G. When the lower bound is reached, the graph is said to be Type 1. In 2018, the direct product of cycle graphs Cm×Cn, for m=3p,5ℓ,8ℓ with p≥2 and ℓ≥1, and arbitrary n≥3, was proved to be Type 1 and suggested the conjecture that, except for C4×C4, the direct product of cycle graphs Cm×Cn with m,n≥3 is Type 1. We prove this conjecture and search further for sufficient conditions to ensure that the direct product of graphs is Type 1. We ask whether one factor being Type 1 is enough to ensure that the direct product also is a Type 1 graph. We prove that the direct product of a conformable regular graph with a regular graph is always conformable. We also prove that the direct product of a Type 1 graph with a bipartite graph is always Type 1.

Efficacy of information extraction from bar, line, circular, bubble and radar graphs

With the emergence of enormous amounts of data, numerous ways to visualize such data have been used. Bar, circular, line, radar and bubble graphs that are ubiquitous were investigated for their effectiveness. Fourteen participants performed four types of evaluations: between categories (cities), within categories (transport modes within a city), all categories, and a direct reading within a category from a graph. The representations were presented in random order and participants were asked to respond to sixteen questions to the best of their ability after visually scanning the related graph. There were two trials on two separate days for each participant. Eye movements were recorded using an eye tracker. Bar and line graphs show superiority over circular and radial graphs in effectiveness, efficiency, and perceived ease of use primarily due to eye saccades. The radar graph had the worst performance. “Vibration-type” fill pattern could be improved by adding colors and symbolic fills. Design guidelines are proposed for the effective representation of data so that the presentation and communication of information are effective.

Induced odd cycle packing number, independent sets, and chromatic number

AbstractThe induced odd cycle packing number of a graph is the maximum integer such that contains an induced subgraph consisting of pairwise vertex‐disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded Vapnik–Chervonenkis (VC) dimension, and linear independence number admit a randomized efficient polynomial‐time approximation scheme for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers and and any ‐vertex graph of induced odd cycle packing number at most returns in time an independent set of whose size is at least with high probability. In addition, we present ‐boundedness results for graphs with bounded odd cycle packing number, and use them to design a quasipolynomial‐time approximation scheme for the independence number only assuming bounded induced odd cycle packing number.

One-relator quotients of right-angled Artin groups

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to right-angled Artin groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element r of a right-angled Artin group G, certain Magnus subgroups embed in the quotient G=G/N(r); that if r=sn has root s in G then the order of s in G is n, and under slightly stronger conditions that the word problem of G is decidable. We also give conditions under which the question of which Magnus subgroups of G embed in G reduces to the same question in the minimal parabolic subgroup of G containing r. In many cases this allows us to characterise Magnus subgroups which embed in G, via a condition on r and the commutation graph of G, and to find further examples of quotients G where the word and conjugacy problems are decidable. We give evidence that situations in which our main theorem applies are not uncommon, by proving that for cycle graphs with a chord Γ, almost all cyclically reduced elements of the right-angled Artin group G(Γ) satisfy the conditions of the theorem.

The Extended Adjacency Indices for Several Types of Graph Operations

Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index <I>EA(G)</I> is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.