Minmax subtree cover problem on cacti

All graph in this paper is simple and connected graph where $V(G)$ is vertex set and $E(G)$ is edge set. Let function $f : V(G)\longrightarrow \{0, 2,..., 2k_v\}$ as vertex labeling and a function $f: E(G)\longrightarrow \{1, 2,..., k_e\}$ as edge labeling where $k=max\{2k_v,k_e\}$ for $k_v,k_e$ are natural number. The weight of vertex $ u,v\in V(G) $ under $f$ is $w(u)=f(u)+ \Sigma_{uv \in E(G)} f(uv)$. In other words, the function $f$ is called local vertex irregular reflexive labeling if every two adjacent vertices has distinct weight and weight of a vertex is defined as the sum of the labels of vertex and the labels of all edges incident this vertex When we assign each vertex of $G$ with a color of the vertex weight $w(uv)$, thus we say the graph G admits a local vertex irregular reflexive coloring. The minimum number of colors produced from local vertex irregular reflexive coloring of graph $G$ is reflexive local irregular chromatic number denoted by $\chi_{lrvs}(G).$ Furthermore, the minimum $k$ required such that $\chi_{lrvs}(G)=\chi(G)$ is called a local reflexive vertex color strength, denoted by \emph{lrvcs}$(G)$. In this paper, we learn about the local vertex irregular reflexive coloring and obtain \emph{lrvcs}$(G)$ of wheel related graphs.

Multi-way graph partition by stochastic probe

Let M be a positive integer, let a1, a2, …, aM be non-negative reals, and put aM+i = ai for i = 1, 2, 3. Further let each of v1, v2, v3 and δ1, δ2, δ3 be 0 or 1, giving 26 possibilities. This note is concerned with the problem of finding bounds for each of the non-trivial cases out of the 26 cyclic sums

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a graph and <em>k</em> be a positive integer. A total <em>k</em>-labeling of <em>G</em> is a map <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,<em>k</em>}. The vertex weight <em>v</em> under the labeling <em>f</em> is denoted by w_<em>f</em>(<em>v</em>) and defined by <em>w</em>_<em>f</em>(<em>v</em>) = <em>f</em>(<em>v</em>) + \sum_{uv \in{E(G)}} {<em>f</em>(<em>uv</em>)}. A total <em>k</em>-labeling of <em>G</em> is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of <em>G</em>, denoted by <em>tvs</em>(<em>G</em>), is the minimum <em>k</em> such that <em>G</em> has a vertex irregular total <em>k</em>-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let <em>G</em> and <em>H</em> be two connected graphs. Let <em>o</em> be a vertex of <em>H</em>. The comb product between <em>G</em> and <em>H</em>, denoted by <em>G </em>\rhd_o <em>H</em>, is a graph obtained by taking one copy of <em>G</em> and |<em>V</em>(<em>G</em>)| copies of <em>H</em> and grafting the i-th copy of <em>H</em> at the vertex <em>o</em> to the i-th vertex of <em>G</em>. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

Let T be a tree on a set V of nodes. The p-th power T p of T is the graph on V such that any two nodes u and w of V are adjacent in T p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T, if any, such that G=T p. Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T, if any, such that G=T p. Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n 3) (respectively, O(n 4)) time. In this paper, we give O(n+m)-time algorithms for both problems.

A subset T⊆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 set of vertices 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.

A 3.4713-approximation algorithm for the capacitated multicast tree routing problem

Self-avoiding walks interacting with a surface are a useful model of polymer adsorption and one can choose to weight the walk according to the number of vertices or the number of edges in the surface. We consider a combined model of self-avoiding walks on the d-dimensional hypercubic lattice with weights associated with the numbers of vertices and edges in the surface and investigate its phase diagram. We also derive inequalities relating the critical points in the edge and vertex weighting schemes.

Exact and heuristic algorithms for the weighted total domination problem

Brain Electroencephalography (EEG) classification is widely applied to analyze cerebral diseases in recent years. Unfortunately, invalid/noisy EEGs degrade the diagnosis performance and most previously developed methods ignore the necessity of EEG selection for classification. To this end, this paper proposes a novel maximum weight clique-based EEG selection approach, named mwcEEGs, to map EEG selection to searching maximum similarity-weighted cliques from an improved Fréchet distance-weighted undirected EEG graph simultaneously considering edge weights and vertex weights. Our mwcEEGs improves the classification performance by selecting intra-clique pairwise similar and inter-clique discriminative EEGs with similarity threshold δ. Experimental results demonstrate the algorithm effectiveness compared with the state-of-the-art time series selection algorithms on real-world EEG datasets.

In this paper we propose to generalize Jaccard and related measures, often used as similarity coefficients between two sets. We define Jaccard, Dice-Sorensen and Tversky edge weights on a graph and generalize them to account for vertex weights. We develop an efficient parallel algorithm for computing Jaccard edge and PageRank vertex weights. We highlight that the weights computation can obtain more than 10X speedup on the GPU versus CPU on large realistic data sets. Also, we show that finding a minimum balanced cut for modified weights can be related to minimizing the sum of ratios of the intersection and union of nodes on the boundary of clusters. Finally, we show that the novel weights can improve the quality of the graph clustering by about 15% and 80% for multi-level and spectral graph partitioning and clustering schemes, respectively.

Given a non-negative integer q, the maximum q-colourable induced subgraph (MCS) problem asks for an induced subgraph of maximum order among those that are q-colourable. By a result of Yannakakis and Gavril and independently of Corneil and Fonlupt, the MCS is NP-Complete even if restricted to the class of split graphs. In this paper, we investigate the approximability of the MCS problem in the class of perfect graphs.

We study the computational complexity of the map redistricting problem (gerrymandering). Mathematically, the electoral district designer (gerrymanderer) attempts to partition a weighted graph into k connected components (districts) such that its candidate (party) wins as many districts as possible. Prior work has principally concerned the special cases where the graph is a path or a tree. Our focus concerns the realistic case where the graph is planar. We prove that the gerrymandering problem is solvable in polynomial time in λ-outerplanar graphs, when the number of candidates and λ are constants and the vertex weights (voting weights) are polynomially bounded. In contrast, the problem is NP-complete in general planar graphs even with just two candidates. This motivates the study of approximation algorithms for gerrymandering planar graphs. However, when the number of candidates is large, we prove it is hard to distinguish between instances where the gerrymanderer cannot win a single district and instances where the gerrymanderer can win at least one district. This immediately implies that the redistricting problem is inapproximable in polynomial time in planar graphs, unless P=NP. This conclusion appears terminal for the design of good approximation algorithms - but it is not. The inapproximability bound can be circumvented as it only applies when the maximum number of districts the gerrymanderer can win is extremely small, say one. Indeed, for a fixed number of candidates, our main result is that there is a constant factor approximation algorithm for redistricting unweighted planar graphs, provided the optimal value is a large enough constant.

The Plane Diameter Completion problem asks, given a plane graph G and a positive integer d, if it is a spanning subgraph of a plane graph H that has diameter at most d. We examine two variants of this problem where the input comes with another parameter k. In the first variant, called BPDC, k upper bounds the total number of edges to be added and in the second, called BFPDC, k upper bounds the number of additional edges per face. We prove that both problems are NP-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n^{3})+2^{2^{O((kd)^2\log d)}} * n steps.

Given n positive integers, the Modular Subset Sum problem asks if a subset adds up to a given target t modulo a given integer m. This is a natural generalization of the Subset Sum problem (where m = +∞) with ties to additive combinatorics and cryptography.Recently, in [Bri17, KX17], efficient algorithms have been developed for the non-modular case, running in near-linear pseudo-polynomial time. For the modular case, however, the best known algorithm by Koiliaris and Xu [KX17] runs in time Õ(m5/4).In this paper, we present an algorithm running in Õ(m) randomized time, which matches a recent conditional lower bound of [ABHS17] based on the Strong Exponential Time Hypothesis. Interestingly, in contrast to most previous results on Subset Sum, our algorithm does not use the Fast Fourier Transform. Instead, it is able to simulate the “textbook” Dynamic Programming algorithm much faster, using ideas from linear sketching. This is one of the first applications of sketching-based techniques to obtain fast algorithms for exact combinatorial problems in an offline setting.