TREE RELATED EXTENDED MEAN CORDIAL GRAPHS

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -- which asks for a subgraph with minimum degree that connects a given set of vertices -- is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.The standard techniques for solving degree-bounded problems often fall in the category of iterative and dependent rounding techniques. Unfortunately, these rounding methods are inherently difficult to adapt to an online settings since the underlying fractional solution may change dramatically in between the rounding steps. Indeed, this might be the very reason that despite many advances in the online network design paradigm in the past two decades, the natural family of degree-bounded problems has remained widely open.In this paper, we design an intuitive greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Furer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach and its dual analysis, paves the way for solving the online variants of the classical problems in this family of problems.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Given a bijection \(f:V(G) \rightarrow \{1,2,\cdots ,|V(G)|\}\), we associate two integers \(S=f(u)+f(v)\) and \(D=|f(u)-f(v)|\) with every edge uv in E(G). The labeling f induces an edge labeling \(f':E(G) \rightarrow \{0,1\}\) such that for any edge uv in E(G), \(f'(uv)=1\) if \(gcd(S,D)=1\), and \(f'(uv)=0\) otherwise. Let \(e_{f'}(i)\) be the number of edges labeled with \(i \in \{0,1\}\). We say f is SD-prime cordial labeling if \(|e_{f'}(0)-e_{f'}(1)| \le 1\). Moreover G is SD-prime cordial if it admits SD-prime cordial labeling. In this paper, we investigate some new construction of SD-prime cordial graph.

Ordered Ramsey numbers

Our aim is to prove some results concerning the connectivity, lineconnectivity and J-connection or connection modulo J of total graphs. Only finite, nonoriented graphs without loops or multiple edges will be considered. Let G be a graph whose vertex set is V(G) and whose edge set is E(G). The elements of the set V(G)wE(G) will be called the elements of G and two elements of G are said to be associated if they are either adjacent or incident. The total graph T(G) of G is a graph whose vertex set is V(G)uE(G), two vertices being joined by an edge if and only if they are associated elements of G (see [1]). As an example of a graph G and its total graph T(G) see Fig. 1. We make an obvious distinction: small rings represent point-vertices of T(G) (x is a point-vertex x ~ E(G)). T(G) contains both G and its line-graph or interchange graph [4] L(G) as disjoint subgraphs. Remember that by definition, V[L(G)] =E(G) and two vertices of L(G) are linked by an edge if and only if the corresponding edges of G are adjacent. Edges of T(G) belonging neither to G nor L(G) form what will be called the incidence-graph I(G) of G. By definition the connectivity k(G) of G is the least number of vertices whose removal disconnects G or reduces G to a single vertex; a set of k(G) vertices satisfying this condition is called a minimal separating vertex set of G. Moreover G is n-connected if and only if k(G) >= n. On the

We want to find the vertex sets of components of a graph G with a known vertex set V and unknown edge set E. We learn about G by sending an oracle a query set $S \subseteq V$\hspace*{-1pt}, and the oracle tells us the vertices connected to S. The objective is to use the minimum number of queries to partition the vertex set into components. The problem is also known as interconnect diagnosis of wiring networks in VLSI. We present a deterministic algorithm using O(min{k, lg n}) queries and a randomized algorithm using expected O(min{k,lg k + lg lg n}) queries, where n is the number of vertices and k is the number of components. We also prove matching lower bounds.

We address synthesis problems in the general context of network reliability, and describe two procedures − CENTERTREE and EXCENTER − whose running times never grow more than cubically with the number of vertices of the network. The first procedure solves two basic related problems − the most linking center and the most linking tree. The first problem requires locating a point on a network through which every connection between two given sets of vertices has to pass. This location maximizes the minimum reliability of the most reliable path connecting through the center every pair of vertices of the two given sets. The second problem asks for a most reliable tree connecting the two sets of vertices. The second procedure generalizes the first one and solves a constrained version of the most linking center problem that involves two (instead of one) reliabilities assigned to each network element and four (instead of two) sets of vertices to be connected. In general, the similarly extended, constrained version of the most linking tree problem is NP-hard, and only in a restricted case is it solvable by procedure EXCENTER. We prove that two extensions of the most linking tree problem, involving edge costs, are NP-hard.

Let us say a graph is Os-free, where s≥1 is an integer, if there do not exist s cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when s=2, is not well understood. For instance, until now we did not know how to test whether a graph is O2-free in polynomial time; and there was an open conjecture, due to Ngoc Khang Le, that O2-free graphs have only a polynomial number of induced paths.In this paper we prove Le's conjecture; indeed, we will show that for all s≥1, there exists c>0 such that every Os-free graph G has at most |G|c induced paths, where |G| is the number of vertices. This provides a poly-time algorithm to test if a graph is Os-free, for all fixed s.The proof has three parts. First, there is a short and beautiful proof, due to Le, that reduces the question to proving the same thing for graphs with no cycles of length four. Second, there is a recent result of Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek, that in every Os-free graph G with no cycle of length four, there is a set of vertices that intersects every cycle, with size logarithmic in |G|. And third, there is an argument that uses the result of Bonamy et al. to deduce the theorem. The last is the main content of this paper.

Minimum face-spanning subgraphs of plane graphs

Let G be a simple connected graph with a set of vertices V(G) and set of edges E(G). The distance from vertex u to vertex v on graph G is the length of the shortest path from vertex u to vertex v, which is denoted by d(u, v). Suppose there is an ordered set W = {w1, w2, w3,…,wn} and v is a vertex on graph G, then the representation of vertex v to W is an n - tuple ordered pair, namely r(v|W) = (d(v, w1),d(v, w2),d(v,w3),…, d(v, wn)). A set W is called a local resolving set if r(u|W) ≠ r(v|W) for each pair of vertex u and vertex v adjacent to each other on graph G. The number of vertices in the local metric base of graph G is called the local metric dimension and is denoted by lmd(G). Dipyramidal graph DPn is the skeleton of an n-sided dipyramid with n ≥ 3. King graph Kmn is a graph with m × n vertices in which each vertex represents a square in an m × n chessboard, and each edge corresponds to a legal move by a king with m, n ≥ 2. In this paper, the local metric dimension of a dipyramidal graph and a king graph were explained.

A graph $\Gamma $ is $G$-symmetric if $\Gamma $ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Gamma $, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\Gamma )$, namely when $V(\Gamma )$ admits a nontrivial $G$-invariant partition ${\mathcal{B}}$, the quotient graph $\Gamma _{\mathcal{B}}$ of $\Gamma $ with respect to ${\mathcal{B}}$ is always $G$-symmetric and sometimes even $(G, 2)$-arc transitive. (A $G$-symmetric graph is $(G, 2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for $\Gamma _{{\mathcal{B}}}$ to be $(G, 2)$-arc transitive (regardless of whether $\Gamma $ is $(G, 2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of ${\mathcal{B}}$ and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v, k$ and two other parameters with respect to $(\Gamma , {\mathcal{B}})$ together with a certain 2-point transitive block design induced by $(\Gamma , {\mathcal{B}})$. We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for $\Gamma _{{\mathcal{B}}}$ to be $(G, 2)$-arc transitive.

Given an integer k ≥ 2, a k-path is a path on k vertices. A set of vertices in a graph G is called a k-path vertex cover if it includes at least one vertex of every k-path of G. A minimum k-path vertex cover in G is a k-path vertex cover having the smallest possible number of vertices and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, the goal is to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number.

Let us denote by Ωn the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff---von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let $\varOmega^{(2)}_{n}$ be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entries in which every line sums to 1. What can be said about $\varOmega ^{(2)}_{n}$'s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains nź1 zeros and a single 1 entry. Indeed, every Latin square of order n is a vertex of $\varOmega^{(2)}_{n}$, but as we show, such vertices constitute only a vanishingly small subset of $\varOmega^{(2)}_{n}$'s vertex set. More concretely, we show that the number of vertices of $\varOmega ^{(2)}_{n}$ is at least $(L_{n})^{\frac{3}{2}-o(1)}$, where Ln is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n×n×n arrays where the entries in every coordinate hyperplane sum to 1, improving a result from Kravtsov (Cybern. Syst. Anal., 43(1):25---33, 2007). Several open questions are presented as well.

Let G = (V, E) be a finite and simple graph with vertex set V (G) and edge set E(G), having order p and size q. A graph G is called super edge-magic if there exists a bijection f : V(G) ∪ E(G) → {1, 2, …, p + q} such that f (V(G)) = {1, 2, …, p} and f (u) + f (uv) + f (v) = k, for every edge uv ∈ E(G). The super edge-magic deficiency of a graph G, denoted by μs(G), is either the minimum nonnegative integer n such that G ∪ nK1 is super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of forests where its components are subdivided stars or paths.

On 3-total edge product cordial labeling of a carbon nanotube network