Upper signed k-domination in a general graph

Let k be a positive integer, and let G=(V,E) be a graph with minimum degree at least k−1. A function f:V→{−1,1} is said to be a signed k-dominating function (SkDF) if _∑u∈N[v]f(u)?k for every v∈V. An SkDF f of a graph G is minimal if there exists no SkDF g such that g≠f and g(v)?f(v) for every v∈V. The maximum of the values of _∑v∈Vf(v), taken over all minimal SkDFs f, is called the upper signed k-domination numberΓkS(G). In this paper, we present a sharp upper bound on this number for a general graph.     

A new approach for approximating node deletion problems

We present a new approach for approximating node deletion problems by combining the local ratio and the greedy multicovering algorithms. For a function , our approach allows to design a 2+max_v∈V(G)logf(v) approximation algorithm for the problem of deleting a minimum number of nodes so that the degree of each node v in the remaining graph is at most f(v). This approximation ratio is shown to be asymptotically optimal. The new method is also used to design a 1+(log2)(k−1) approximation algorithm for the problem of deleting a minimum number of nodes so that the remaining graph contains no k-bicliques.     

Group path covering and distance two labeling of graphs

For a positive integer d, an L(d,1)-labeling f of a graph G is an assignment of integers to the vertices of G such that |f(u)−f(v)|?d if uv∈E(G), and |f(u)−f(v)|?1 if u and u are at distance two. The span of an L(d,1)-labeling f of a graph is the absolute difference between the maximum and minimum integers used by f. The L(d,1)-labeling number of G, denoted by λd,1(G), is the minimum span over all L(d,1)-labelings of G. An L^′(d,1)-labeling of a graph G is an L(d,1)-labeling of G which assigns different labels to different vertices. Denote by the L^′(d,1)-labeling number of G. Georges et al. [Discrete Math. 135 (1994) 103-111] established relationship between the L(2,1)-labeling number of a graph G and the path covering number of Gc, the complement of G. In this paper we first generalize the concept of the path covering of a graph to the t-group path covering. Then we establish the relationship between the L^′(d,1)-labeling number of a graph G and the (d−1)-group path covering number of Gc. Using this result, we prove that and for bipartite graphs G can be computed in polynomial time.     

Weighted Coloring: further complexity and approximability results

Given a vertex-weighted graph G=(V,E;w), w(v)?0 for any v∈V, we consider a weighted version of the coloring problem which consists in finding a partition S=(S₁,…,Sk) of the vertex set of G into stable sets and minimizing where the weight of S is defined as . In this paper, we continue the investigation of the complexity and the approximability of this problem by answering some of the questions raised by Guan and Zhu [D.J. Guan, X. Zhu, A coloring problem for weighted graphs, Inform. Process. Lett. 61 (2) (1997) 77-81].     

Ranking numbers of graphs

Given a graph G, a vertex ranking (or simply, ranking) of G is a mapping f from V(G) to the set of all positive integers, such that for any path between two distinct vertices u and v with f(u)=f(v), there is a vertex w in the path with f(w)>f(u). If f is a ranking of G, the ranking number of G under f, denoted γf(G), is defined by , and the ranking number of G, denoted γ(G), is defined by . The vertex ranking problem is to determine the ranking number γ(G) of a given graph G. This problem is a natural model for the manufacturing scheduling problem. We study the ranking numbers of graphs in this paper. We consider the relation between the ranking numbers and the minimal cut sets, and the relation between the ranking numbers and the independent sets. From this, we obtain the ranking numbers of the powers of paths and the powers of cycles, the Cartesian product of P₂ with Pn or Cn, and the caterpilars. And we also find the vertex ranking numbers of the composition of two graphs in this paper.     

Vertex vulnerability parameters of Kronecker products of complete graphs

Let G₁ and G₂ be two graphs. The Kronecker product G₁×G₂ of G₁ and G₂ has vertex set V(G₁×G₂)=V(G₁)×V(G₂) and edge set and v₁v₂∈E(G₂)}. In this paper, we determine some vertex vulnerability parameters of the Kronecker product of complete graphs Km×Kn for n?m?2 and n?3.     

The edge-orientation problem and some of its variants on weighted graphs

Let G(V, E) be a connected undirected graph with n vertices and m edges, where each vertex v is associated with a cost C(v) and each edge e = (u, v) is associated with two weights, W(u → v) and W(v → u). The issue of assigning an orientation to each edge so that G becomes a directed graph is resolved in this paper. Determining a scheme to assign orientations of all edges such that max_x∈V{C(x)+∑_x→zW(x→z)} is minimized is the objective. This issue is called the edge-orientation problem (the EOP). Two variants of the EOP, the Out-Degree-EOP and the Vertex-Weighted EOP, are first proposed and then efficient algorithms for solving them on general graphs are designed. Ascertaining that the EOP is NP-hard on bipartite graphs and chordal graphs is the second result. Finally, an O(n log n)-time algorithm for the EOP on trees is designed. In general, the algorithmic results in this paper facilitate the implementation of the weighted fair queuing (WFQ) on real networks. The objective of the WFQ is to assign an effective weight for each flow to enhance link utilization. Our findings consequently can be easily extended to other classes of graphs, such as cactus graphs, block graphs, and interval graphs.     

Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes

An adjacent vertex-distinguishing edge coloring of a simple graph G is a proper edge coloring of G such that incident edge sets of any two adjacent vertices are assigned different sets of colors. A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G=(V,E) is a proper total coloring of G such that H(u)≠H(v) for any two adjacent vertices u and v, where H(u)={h(wu)|wu∈E(G)}∪{h(u)} and H(v)={h(xv)|xv∈E(G)}∪{h(v)}. The minimum number of colors required for an adjacent vertex-distinguishing edge coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the adjacent vertex-distinguishing edge chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by (resp. χat(G)). In this paper, we consider the adjacent vertex-distinguishing edge chromatic number and adjacent vertex-distinguishing total chromatic number of the hypercube Qn, prove that for n?3 and χat(Qn)=Δ(Qn)+2 for n?2.     

Enhanced algorithms for Local Search

Let G=(V,E) be a finite graph, and be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity , so that we obtain a deterministic query complexity of , where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of . Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Note on the connectivity of line graphs

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G).A subset S of E(G) is called a restricted edge-cut if G−S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ₂(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ₂(G)?λ(G)?κ(G).In 1969, Chartrand and Stewart have shown that

     

On the profile of the corona of two graphs

The concept of profile, together with bandwidth, originates from handling sparse matrices in solving linear systems of equations. Given a graph G, the profile minimization problem is to find a one-to-one mapping f:V(G)→{1,2,…,|V(G)|} such that ∑_v∈V(G)max_x∈N[v](f(v)−f(x)) is as small as possible, where N[v]={v}∪{x:x is adjacent to v}. This paper studies the profile of the corona G∧H of two graphs G and H. In particular, bounds for the profile of the corona of two graphs are established. Also, exact values of the profiles of coronas G∧H are obtained when G has certain properties, including when G is a caterpillar, a complete graph or a cycle.     

Consensus models: Computational complexity aspects in modern approaches to the list coloring problem

In the paper we study new approaches to the problem of list coloring of graphs. In the problem we are given a simple graph G=(V,E) and, for every v∈V, a nonempty set of integers S(v); we ask if there is a coloring c of G such that c(v)∈S(v) for every v∈V. Modern approaches, connected with applications, change the question—we now ask if S can be changed, using only some elementary transformations, to ensure that there is such a coloring and, if the answer is yes, what is the minimal number of changes. In the paper for studying the adding, the trading and the exchange models of list coloring, we use the following transformations:

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adding of colors (the adding model): select two vertices u, v and a color c∈S(u); add c to S(v), i.e. set S(v):=S(v)∪{c};

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trading of colors (the trading model): select two vertices u, v and a color c∈S(u); move c from S(u) to S(v), i.e. set S(u):=S(u)?{c} and S(v):=S(v)∪{c};

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exchange of colors (the exchange model): select two vertices u, v and two colors c∈S(u), d∈S(v); exchange c with d, i.e. set S(u):=(S(u)?{c})∪{d} and S(v):=(S(v)?{d})∪{c}.

Our study focuses on computational complexity of the above models and their edge versions. We consider these problems on complete graphs, graphs with bounded cyclicity and partial k-trees, receiving in all cases polynomial algorithms or proofs of NP-hardness.     

On the complexity of signed and minus total domination in graphs

In this paper we present unified methods to solve the minus and signed total domination problems for chordal bipartite graphs and trees in O(n²) and O(n+m) time, respectively. We also prove that the decision problem for the signed total domination problem on doubly chordal graphs is NP-complete. Note that bipartite permutation graphs, biconvex bipartite graphs, and convex bipartite graphs are subclasses of chordal bipartite graphs.