Two-factors in orientated graphs with forbidden transitions

The instance of the problem of finding 2-factors in an orientated graph with forbidden transitions consists of an orientated graph G and for each vertex v of G, a graph H_v of allowed transitions at v. Vertices of the graph H_v are the edges incident to v. An orientated 2-factor F of G is called legal if all the transitions are allowed, i.e. for every vertex v, the two edges of F adjacent to it form an edge in H_v. Deciding whether a legal 2-factor exists in G is NP-complete in general. We investigate the case when the graphs of allowed transitions are taken from some fixed class C. We provide an exact characterization of classes C so that the problem is NP-complete. In particular, we prove a dichotomy for this problem, i.e. that for any class C it is either polynomial or NP-complete.     

PTAS for the minimum k-path connected vertex cover problem in unit disk graphs

In the Minimum k-Path Connected Vertex Cover Problem (MkPCVCP), we are given a connected graph G and an integer k ≥ 2, and are required to find a subset C of vertices with minimum cardinality such that each path with length k ? 1 has a vertex in C, and moreover, the induced subgraph G[C] is connected. MkPCVCP is a generalization of the minimum connected vertex cover problem and has applications in many areas such as security communications in wireless sensor networks. MkPCVCP is proved to be NP-complete. In this paper, we give the first polynomial time approximation scheme (PTAS) for MkPCVCP in unit disk graphs, for every fixed k ≥ 2.     

Comparing Zagreb indices for connected graphs

It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M₂(G)/m≥M₁(G)/n, where M₁ and M₂ are the first and second Zagreb indices. Hansen and Vuki?evi? proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m−n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M₂/(m−1)>M₁/n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.     

Panconnectivity for interconnection networks with faulty elements

Let Go and G1 be two graphs with the same vertices. The new graph G（G0, G1; M） is a graph with the vertex set V（0o） ∪）V（G1） and the edge set E（Go） UE（G1） UM, where M is an arbitrary perfect matching between the vertices of Go and G1, i.e., a set of cross edges with one endvertex in Go and the other endvertex in G1. In this paper, we will show that if Go and G1 are f-fault q-panconnected, then for any f 〉 2, G（G0, G1; M） is （f ＋ 1）-fault （q ＋ 2）-panconnected.     

splitting number is NP-complete

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k⩾0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v₁ and v₂, and attaches the neighbors of v either to v₁ or to v₂. We prove that the splitting number decision problem is NP-complete when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs implies NP-completeness for graphs not containing a subdivision of K₅ as a subgraph.     

On the 2-rainbow domination in graphs

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G□K₂. In this paper, we show that the problem of deciding if a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs. Exact values of 2-rainbow domination numbers of several classes of graphs are found, and it is shown that for the generalized Petersen graphs GP(n,k) this number is between ⌈4n/5⌉ and n with both bounds being sharp.     

V-shop scheduling

This paper deals with V-shop scheduling where the route by which a job passes through the machines is: M₁ → M₂ → ? → M_m?1 → M_m → M_m?1 → ? → M₂ → M₁. Flowshop scheduling is a special case of V-shop. The two-machine, minimum schedule length V-shop scheduling problem is proved to be binary NP-complete. Efficient solvable algorithms are presented for some simple special cases of NP-complete V-shop problems; specifically n/2/V/C_max with a dominating machine and n/m/V, t_ij = 1/(C_max ∩ ΣC_i). n/m/V/C_max, m?2, with an increasing series of dominating machines is discussed.     

The chain graph sandwich problem

The chain graph sandwich problem asks: Given a vertex set V, a mandatory edge set E ¹, and a larger edge set E ², is there a graph G=(V,E) such that E ¹?E?E ² with G being a chain graph (i.e., a 2K ₂-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E ¹ is a connected graph, which has a linear-time solution, (2) the threshold graph sandwich problem, which has a linear-time solution, and (3) the chain probe graph problem, which has a polynomial-time solution.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid

Given an undirected graph G=(V,E) with matching number ν(G), a d-blocker is a subset of edges B such that ν((V,E?B))≤ν(G)−d and a d-transversal T is a subset of edges such that every maximum matching M has |M∩T|≥d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the special cases where G is a grid graph or a tree.     

The complexity of generalized clique packing

The K_i - j packing problem P_{i, j} is defined as follows: Given a graph G and integer k does there exist a set of at least kK_i's in G such that no two of these K_i's intersect in more than j nodes. This problem includes such problems as matching, vertex partitioning into complete subgraphs and edge partitioning into complete subgraphs. In this paper it is shown thhat for i ? 3 and 0?j?i ?2 the P_{i, j} problems is NP-complete. Furthermore, the problems remains NP-complete for i?3 and 1?j?i ?2 for chordal graphs.     

Ki-covers I: Complexity and polytopes

A K_i in a graph is a complete subgraph of size i. A K_i-cover of a graph G(V, E is a set C of K_{i − 1}'s of G such that every K_i in G contains at least one K_{i − 1} in C. Thus a K₂-cover is a vertex cover. The problem of determining whether a graph has a K_i-cover (i ⩾ 2) of cardinality ⩽k is shown to be NP-complete for graphs in general. For chordal graphs with fixed maximum clique size, the problem is polynomial; however, it is NP-complete for arbitrary chordal graphs when i ⩾ 3. The NP-completeness results motivate the examination of some facets of the corresponding polytope. In particular we show that various induced subgraphs of G define facets of the K_i-cover polytope. Further results of this type are also produced for the K₃-cover polytope. We conclude by describing polynomial algorithms for solving the separation problem for some classes of facets of the K_i-cover polytope.     

One-basedness and groups of the form G/G 00

We initiate a geometric stability study of groups of the form G/G ⁰⁰, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M?? with a predicate for G ⁰⁰ and check 1-basedness or non-1-basedness for G/G ⁰⁰, where G is an additive truncation of M, a multiplicative truncation of M, SO ₂(M) or one of its truncations; such groups G/G ⁰⁰ are now interpretable in M??. We prove that the only 1-based groups are those where G is a sufficiently ??big?? multiplicative truncation, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-hyperdefinable groups constructed from those above.     

Finite Abelian actions on surfaces

Edmonds showed that two free orientation preserving smooth actions φ₁ and φ₂ of a finite Abelian group G on a closed connected oriented smooth surface M are equivalent by an equivariant orientation preserving diffeomorphism iff they have the same bordism class [M,φ₁]=[M,φ₂] in the oriented bordism group Ω₂(G) of the group G. In this paper, we compute the bordism class [M,φ] for any such action of G on M and we determine for a given M, the bordism classes in Ω₂(G) that are representable by such actions of G on M. This will enable us to obtain a formula for the number of inequivalent such actions of G on M. We also determine the “weak” equivalence classes of such actions of G on M when all the p-Sylow subgroups of G are homocyclic (i.e. of the form n(Z/pαZ)).     

Enumerating disjunctions and conjunctions of paths and cuts in reliability theory

Let G=(V,E) be a (directed) graph with vertex set V and edge (arc) set E. Given a set P of source-sink pairs of vertices of G, an important problem that arises in the computation of network reliability is the enumeration of minimal subsets of edges (arcs) that connect/disconnect all/at least one of the given source-sink pairs of P. For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. Furthermore, under the assumption that P consists of all pairs within a given vertex set, we also give incremental polynomial time algorithm for enumerating all minimal path disjunctions and cut conjunctions. For directed graphs, the enumeration problem for cut disjunction is known to be NP-complete. We extend this result to path conjunctions and path disjunctions, leaving open the complexity of the enumeration of cut conjunctions. Finally, we give a polynomial delay algorithm for enumerating all minimal sets of arcs connecting two given nodes s₁ and s₂ to, respectively, a given vertex t₁, and each vertex of a given subset of vertices T₂.     

Searching for an edge in a graph with restricted test sets

Consider the (2,n) group testing problem with test sets of cardinality at most 2. We determine the worst case number c₂ of tests for this restricted group testing problem.Furthermore, using a game theory approach we solve the generalization of this group testing problem to the following search problem, which was suggested by Aigner in [M. Aigner, Combinatorial Search, Wiley-Teubner, 1988]: Suppose a graph G(V,E) contains one defective edge e. We search for the endpoints of e by asking questions of the form “Is at least one of the vertices of X an endpoint of e?”, where X is a subset of V with |X|≤2. What is the minimum number c₂(G) of questions, which are needed in the worst case to identify e?We derive sharp upper and lower bounds for c₂(G). We also show that the determination of c₂(G) is an NP-complete problem. Moreover, we establish some results on c₂ for random graphs.     

The evolution of the min-min random graph process

We study the following min-min random graph process G=(G₀,G₁,…): the initial state G₀ is an empty graph on n vertices (n even). Further, G_M+1 is obtained from G_M by choosing a pair {v,w} of distinct vertices of minimum degree uniformly at random among all such pairs in G_M and adding the edge {v,w}. The process may produce multiple edges. We show that G_M is asymptotically almost surely disconnected if M≤n, and that for M=(1+t)n, constant, the probability that G_M is connected increases from 0 to 1. Furthermore, we investigate the number X of vertices outside the giant component of G_M for M=(1+t)n. For constant we derive the precise limiting distribution of X. In addition, for n⁻¹ln⁴n≤t=o(1) we show that tX converges to a gamma distribution.     

Minimal codes in abelian group algebras

In this paper we show that two minimal codes $\text{M}$₁ and $\text{M}$₂ in the group algebra $\text{F}$₂[G] have the same (Hamming) weight distribution if and only if there exists an automorphism θ of G whose linear extension to $\text{F}$₂[G] maps $\text{M}$₁ onto $\text{M}$₂. If θ(M₁) = M₂, then $\text{M}$₁ and $\text{M}$₂ are called equivalent. We also show that there are exactly τ(l) inequivalent minimal codes in $\text{F}$₂[G], where ? is the exponent of G, and τ(?) is the number of divisors of ?.     

The parameterized complexity of the induced matching problem

Given a graph G and an integer k≥0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs, as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.