Embedding Every Cycle in Conditionally Faulty Folded Hypercubes

The folded hypercube is a well-known variation of hypercube structure and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let [Formula: see text] (respectively, [Formula: see text]) denote the set of faulty vertices (respectively, faulty edges) in an [Formula: see text]-dimensional folded hypercube [Formula: see text]. In the case that all edges in [Formula: see text] are fault-free, Cheng et al. [Cycles embedding on folded hypercubes with faulty vertices, Discrete Appl. Math. 161 (2013) 2894–2900] has shown that (1) every fault-free edge of [Formula: see text] lies on a fault-free cycle of every even length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text]; and (2) every fault-free edge of [Formula: see text] lies on a fault-free cycle of every odd length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text] is even. In this paper, we extend Cheng’s result to obtain two further properties, which consider both vertex and edge faults, as follows: (1) Every fault-free edge of [Formula: see text] lies on a fault-free cycle of every even length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text]; (2) Every fault-free edge of [Formula: see text] lies on a fault-free cycle of every odd length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text] is even.

Fault-tolerant cycles embedding in hypercubes with faulty edges

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

Hamiltonian cycles in hypercubes with faulty edges

On the mutually independent Hamiltonian cycles in faulty hypercubes

Vertex-fault-tolerant cycles embedding in balanced hypercubes

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm × Cn with n even is strongly hamiltonian-laceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn - 2 unless both faulty elements are contained in the same partite vertex set; otherwise, it has a fault-free cycle of length mn - 4. A sufficient condition is derived for the graph with two faulty edges to have a hamiltonian cycle. By applying fault-hamiltonicity of Pm × Cn to a two-dimensional torus Cm × Cn, we obtain interesting hamiltonian properties of a faulty Cm × Cn.

Conditional edge-fault pancyclicity of augmented cubes

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.

An optimal result on fault-tolerant cycle-embedding in alternating group graphs

On isometric subgraphs of infinite bridged graphs and geodesic convexity

A graph is denoted by $G$ with the vertex set $V(G)$ and the edge set $E(G)$.A path $P=\langle v_0,v_1,\cdots,v_m\rangle$ is a sequence of adjacent vertices. Two paths with equal length $P_1=\langle u_1,u_2,\ldots,u_m\rangle$ and $P_2=\langle v_1,v_2,\ldots,v_m\rangle$ from $a$ to $b$ are {\it independent} if $u_1=v_1=a$, $u_m=v_m=b$, and $u_i\neq v_i$ for $2\leq i\leq m-1$.Paths with equal length $\{P_i\}^{n}_{i=1}$ from $a$ to $b$ are {\it mutually independent} if they are pairwisely independent. Let $u$ and $v$ be two distinct vertices of a bipartite graph $G$, and let $l$ be a positive integer length, $d_G(u,v)\le l \le |V(G)-1|$ with $(l-d_G(u,v))$ being even. We say that the pair of vertices $u$, $v$ is {\it $(m,l)$-mutually independent bipanconnected} if there exist $m$ mutually independent paths $\{P_i^l\}_{i=1}^m$ with length $l$ from $u$ to $v$. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices $u$ and $v$ in the $n$-dimensional hypercube, with $d_{Q_n}(u,v) \ge n-1$, is $(n-1,l)$-mutually independent bipanconnected for every $l$, $d_{Q_n}(u,v) \le l \le |V(Q_n)-1|$ with $(l-d_{Q_n}(u,v))$ being even. As for $d_{Q_n}(u,v) \le n-2$, it is also $(n-1,l)$-mutually independent bipanconnected if $l \ge d_{Q_n}(u,v)+2$, and is only $(l,l)$-mutually independent bipanconnected if $l=d_{Q_n}(u,v)$.

For an undirected graph $$G=(V,E)$$G=(V,E), a vertex $$\tau \in V$$źźV separates vertices u and v (where $$u,v\in V$$u,vźV, $$u\ne v$$uźv) if their distances to $$\tau $$ź are not equal. Given an integer parameter $$k \ge 1$$kź1, a set of vertices $$L\subseteq V$$L⊆V is a feasible solution, if for every pair of distinct vertices, u, v, there are at least k distinct vertices $$\tau _{1},\tau _{2},\ldots ,\tau _{k}\in L$$ź1,ź2,ź,źkźL, each separating u and v. Such a feasible solution is called a landmark set, and the k-metric dimension of a graph is the minimal cardinality of a landmark set for the parameter k. The case $$k=1$$k=1 is a classic problem, where in its weighted version, each vertex v has a non-negative cost, and the goal is to find a landmark set with minimal total cost. We generalize the problem for $$k \ge 2$$kź2, introducing two models, and we seek for solutions to both the weighted version and the unweighted version of this more general problem. In the model of all-pairs (AP), k separations are needed for every pair of distinct vertices of V, while in the non-landmarks model (NL), such separations are required only for pairs of distinct vertices in $$V \setminus L$$V\L. We study the weighted and unweighted versions for both models (AP and NL), for path graphs, complete graphs, complete bipartite graphs, and complete wheel graphs, for all values of $$k \ge 2$$kź2. We present algorithms for these cases, thus demonstrating the difference between the two new models, and the differences between the cases $$k=1$$k=1 and $$k \ge 2$$kź2.

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.

The k-edge-connectivity augmentation problem for a specified set of vertices of a graph with degree constraints, kECA-SV-DC, is defined as follows: Given an undirected multigraph G = (V, E), a specified set of vertices S ⊆ V and a function g: V→Z+∪{∞}, find a smallest set E' of edges such that (V, E ∪ E') has at least k edge-disjoint paths between any pair of vertices in S and such that, for any νe V, E' includes at most g(v) edges incident to v, where Z+ is the set of nonnegative integers. This paper first shows polynomial time solvability of kECA-SV-DC and then gives a linear time algorithm for 2ECA-SV-DC.