Edges Ab Research Articles

On Total Edge Irregularity Strength of Some Graphs Related to Double Fan Graphs

Let <em>G = (V(G),E(G))</em> be a simple, connected, undirected graph with non empty vertex set <em>V(G)</em> and edge set<em> E(G)</em>. The function <em>f : V(G) ∪ E(G) ↦ </em>{1,2, ...,k} (for some positive integer k) is called an edge irregular total <em>k</em>−labeling where each two edges <em>ab</em> and<em> cd</em>, having distinct weights, that are<em> f (a)+ f (ab)+ f (b) ≠ f (c)+ f (cd)+ f (d).</em> The minimum <em>k</em> for which <em>G</em> has an edge irregular total <em>k</em>−labeling is denoted by tes<em>(G)</em> and called total edge irregularity strength of graph <em>G</em>. In this paper, we determine the exact value of the total edge irregularity strength of double fan ladder graph, centralized double fan graph, and generalized parachute graph with upper path.

Investigation of Disorder in Mixed Phase, sp²-sp³ Bonded Graphene-Like Nanocarbon.

Disorder in a mixed phase, sp2-sp3 bonded graphene-like nanocarbon (GNC) lattice has been extensively studied for its electronic and field emission properties. Morphological investigations are performed using scanning electron microscopy (SEM) which depicts microstructures comprising of atomically flat terraces (c-planes) with an abundance of edges (ab planes which are orthogonal to c-planes). Scanning tunneling microscopy (STM) is used to observe the atomic structure of basal planes whereas field emission microscopy (FEM) is found to be suitable for resolving nanotopography of edges. STM images revealed the hexagonal and non-hexagonal atomic arrangements in addition to a variety of defect structures. Scanning tunneling spectroscopy is carried out to study the effect of this short-range disorder on the local density of states. Current versus voltage (I-V) characteristics have been recorded at different defect sites and are compared with respect to the extent of the defect. As sharp edges of GNC are expected to be excellent field emitters, because of low work function and high electric field, enhancement in current is observed particularly when applied electric field is along basal planes. Therefore, it is worthwhile to investigate field emission from these samples. The FEM images show a cluster of bright spots at low voltages which later transformed into an array resembling ledges of ab-planes with increasing voltage. Reproducible I-V curves yield linear Fowler-Nordheim plots supporting field emission as the dominant mechanism of electron emission. Turn on field for 10 μA current is estimated to be ~3 V/μm.

Maximum weight stable set in ([formula omitted], bull)-free graphs and ([formula omitted], bull)-free graphs

We give a polynomial time algorithm that finds the maximum weight stable set in a graph that does not contain an induced path on seven vertices or a bull (the graph with vertices a,b,c,d,e and edges ab,bc,cd,be,ce). With the same arguments we also give a polynomial algorithm for any graph that does not contain S1,2,3 or a bull.

4-coloring [formula omitted]-free graphs

We present a polynomial-time algorithm that determines whether a graph that contains no induced path on six vertices and no bull (the graph with vertices a,b,c,d,e and edges ab,bc,cd,be,ce) is 4-colorable. We also show that for any fixed k the k-coloring problem can be solved in polynomial time in the class of (P6,bull,gem)-free graphs.

A note on efficient domination in a superclass of [formula omitted]-free graphs

In a graph G=(V,E), an efficient dominating set is a subset D⊆V such that D is an independent set such that each vertex outside D has exactly one neighbor in D. E stands for the graph with vertices a,b,c,d,e,f and edges ab,bc,cd,de and cf, while xNet stands for the graph with vertices a,b,c,d,e,f,g and edges ad,be,cf,de,ef,fd and cg. This note shows that, given an (E,xNet)-free graph, a minimum efficient dominating set (if any) can be found in polynomial time, extending two polynomially solvable cases for efficient dominating sets recently found by Milanič (2013) [4], and Brandstädt et al. (2013) [2].

Maximum weight independent sets in ([formula omitted])-free graphs

The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated problem on graphs, it is well known to be NP-complete and hard to approximate. Several graph classes for which MWIS can be solved in polynomial time have been introduced in the literature. This note shows that MWIS can be solved in polynomial time for (P6,co-banner)-free graphs – where a P6 is an induced path of 6 vertices and a co-banner is a graph with vertices a,b,c,d,e and edges ab,bc,cd,ce,de – so extending different analogous known results for other graph classes, namely, P4-free, 2K2-free, (P5,co-banner)-free, and (P6,triangle)-free graphs. The solution algorithm is based on an idea/algorithm of Farber (1989) [10], leading to a dynamic programming approach for MWIS, and needs none of the aforementioned known results as sub-procedure.

Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Clique separator decomposition, introduced by Whitesides and Tarjan, is one of the most important graph decompositions. A hole is a chordless cycle with at least five vertices. A paraglider is a graph with five vertices a,b,c,d,e and edges ab,ac,bc,bd,cd,ae,de. We show that every (hole, paraglider)-free graph admits a clique separator decomposition into graphs of three very specific types. This yields efficient algorithms for various optimization problems in this class of graphs.

Which claw-free graphs are strongly perfect?

A set S of pairwise nonadjacent vertices in an undirected graph G is called a stable transversal of G if S meets every maximal (with respect to set-inclusion) clique of G . G is called strongly perfect if all its induced subgraphs (including G itself) have stable transversals. A claw is a graph consisting of vertices a , b , c , d and edges ab , ac , ad . We characterize claw-free strongly perfect graphs by five infinite families of forbidden induced subgraphs. This result—whose validity had been conjectured by Ravindra [Research problems, Discrete Math. 80 (1990) 105–107]—subsumes the characterization of strongly perfect line-graphs that was discovered earlier by Ravindra [Strongly perfect line graphs and total graphs, Finite and Infinite Sets. Colloq. Math. Soc. János Bolyai 37 (1981) 621–633].

On the structure of ( [formula omitted],gem)-free graphs

We give a complete structure description of ( P 5 ,gem)-free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P 5 is the induced path with five vertices a , b , c , d , e and four edges ab , bc , cd , de , and the gem consists of a P 4 a , b , c , d with edges ab , bc , cd plus a universal vertex e adjacent to a , b , c , d .

New Graph Classes of Bounded Clique-Width

The clique-width of graphs is a major new concept with respect to the efficiency of graph algorithms; it is known that every problem expressible in a certain kind of Monadic Second Order Logic called LinEMSOL(τ1,L) by Courcelle et al. is linear-time solvable on any graph class with bounded clique-width for which a k-expression for the input graph can be constructed in linear time. The notion of clique-width extends the one of treewidth since bounded treewidth implies bounded clique-width. We give a complete classification of all graph classes defined by forbidden one-vertex extensions of the P4 (i.e., the path with four vertices a,b,c,d and three edges ab,bc,cd) with respect to bounded clique-width. Our results extend and improve recent structural and complexity results in a systematic way.

Optimizing Bull-Free Perfect Graphs

A bull is a graph with five vertices a,b,c,d,e and five edges ab, ac, bc, da, eb. Here we present polynomial-time combinatorial algorithms for the optimal weighted coloring and weighted clique problems in bull-free perfect graphs. The algorithms are based on a structural analysis and decomposition of bull-free perfect graphs.

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Modular decomposition of graphs is a powerful tool for designing efficient algorithms for problems on graphs such as Maximum Weight Stable Set (MWS) and Maximum Weight Clique. Using this tool we obtain O( n· m) time algorithms for MWS on chair- and xbull-free graphs which considerably extend an earlier result on bull- and chair-free graphs by De Simone and Sassano (the chair is the graph with vertices a, b, c, d, e and edges ab, bc, cd, be, and the xbull is the graph with vertices a, b, c, d, e, f and edges ab, bc, cd, de, bf, cf). Moreover, our algorithm is robust in the sense that we do not have to check in advance whether the input graphs are indeed chair- and xbull-free.

( P5,diamond)-free graphs revisited: structure and linear time optimization

Using the concept of prime graphs and modular decomposition of graphs, we give a complete structure description of ( P 5,diamond)-free graphs implying that these graphs have bounded clique width (the P 5 is the induced path with five vertices a, b, c, d, e and four edges ab, bc, cd, de, and the diamond consists of four vertices a, b, c, d such that a, b, c form an induced path with edges ab, bc, and vertex d is adjacent to a, b and c). The structure and bounded clique width of this graph class allows to solve several algorithmic problems on this class in linear time, among them the problems Maximum Weight Stable Set (MWS), Maximum Weight Clique, Domination, Steiner Tree and in general every algorithmic problem which is, roughly speaking, expressible in a certain kind of Monadic Second-Order Logic using quantification only over vertex but not over edge set predicates. This improves previous results on ( P 5,diamond)-free graphs in several ways: We give a complete structure description of prime ( P 5,diamond)-free graphs, we do not only solve the MWS problem on this class, we achieve linear time algorithms (instead of a recent time bound O ( n m ) ), and we can do all this on a larger graph class containing ( P 5,diamond)-free graphs which admits linear time recognition.

Structure and stability number of chair-, co-P- and gem-free graphs revisited

The P 4 is the induced path with vertices a, b, c, d and edges ab, bc, cd. The chair (co-P, gem) has a fifth vertex adjacent to b ( a and b, a, b, c and d, respectively). We give a complete structure description of prime chair-, co-P- and gem-free graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences; very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair-, co-P- and gem-free graphs and a slightly larger class of graphs. This drastically improves a recently published O( n 4) time bound for Maximum Stable Set on butterfly-, chair-, co-P- and gem-free graphs.

Splitting and contractible edges in 4-connected graphs

Let G be a graph and let x be a vertex of degree four with N G ( x)={ a, b, c, d}. Then the operation of deleting x and adding the edges ab and cd is called splitting at x. An edge e of a graph G is said to be k-contractible if contraction of e yields a k-connected graph. Splitting has been studied as a reduction method to preserve edge-connectivity. In this paper, we consider splitting and 4-contractible edges as tools for reduction of 4-connected graphs. We prove that for a 4-connected graph G of order at least six, there exists either a 4-contractible edge or a vertex eligible for splitting preserving 4-connectedness near every vertex in G.

On the recognition of P4-indifferent graphs

A simple graph is P 4 -indifferent if it admits a total order < on its nodes such that every chordless path with nodes a, b, c, d and edges ab, bc, cd has a< b< c< d or a> b> c> d. P 4-indifferent graphs generalize indifferent graphs and are perfectly orderable. Recently, Hoàng; Maffray and Noy gave a characterization of P 4-indifferent graphs in terms of forbidden induced subgraphs. We clarify their proof and describe a linear time algorithm to recognize P 4-indifferent graphs. When the input is a P 4-indifferent graph, then the algorithm computes an order < as above.

New classes of perfectly orderable graphs

This paper generalizes previous works on perfectly orderable graphs by Olariu (Discrete Math. 113 (1992) 143) and by Hoàng et al. (Discrete Math. 102 (1992) 67). Chvátal defined a graph to be perfectly orderable (V. Chvátal, in: C. Berge, V. Chvátal (Eds.), Topics on Perfect Graphs, Annals of Discrete Mathematics, Vol. 21, North-Holland, Amsterdam, 1984, pp. 63–65) if there exists a linear order < on its set of vertices such that no induced path abcd with edges ab, bc, cd has both a< b and d< c. Given a graph G and a vertex v in G such that G− v is perfectly orderable, we set some conditions on v for which we deduce that G is perfectly orderable. Our method allows to construct a new class of such graphs, recognizable in polynomial time, containing quasi-brittle graphs, charming graphs and some other classes of perfectly orderable graphs.

Linear time recognition of P4-indifference graphs

A graph is a P4-indifference graph if it admits an ordering &lt; on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has a

Euler circuits and DNA sequencing by hybridization

Sequencing by hybridization is a method of reconstructing a long DNA string — that is, figuring out its nucleotide sequence — from knowledge of its short substrings. Unique reconstruction is not always possible, and the goal of this paper is to study the number of reconstructions of a random string. For a given string, the number of reconstructions is determined by the pattern of repeated substrings; in an appropriate limit substrings will occur at most twice, so the pattern of repeats is given by a pairing: a string of length 2 n in which each symbol occurs twice. A pairing induces a 2-in, 2-out graph, whose directed edges are defined by successive symbols of the pairing — for example the pairing ABBCAC induces the graph with edges AB, BB, BC, and so forth — and the number of reconstructions is simply the number of Euler circuits in this 2-in, 2-out graph. The original problem is thus transformed into one about pairings: to find the number f k ( n) of n-symbol pairings having k Euler circuits. We show how to compute this function, in closed form, for any fixed k, and we present the functions explicitly for k=1,…,9. The key is a decomposition theorem: the Euler “circuit number” of a pairing is the product of the circuit numbers of “component” sub-pairings. These components come from connected components of the “interlace graph”, which has the pairing's symbols as vertices, and edges when symbols are “interlaced”. ( A and B are interlaced if the pairing has the form ⋯ A⋯ B⋯ A⋯ B⋯ or ⋯ B⋯ A⋯ B⋯ A⋯.) We carry these results back to the original question about DNA strings, and provide a total variation distance upper bound for the approximation error. We perform an asymptotic enumeration of 2-in, 2-out digraphs to show that, for a typical random n-pairing, the number of Euler circuits is of order no smaller than 2 n / n, and the expected number is asymptotically at least e −1/22 n−1 / n. Since any n-pairing has at most 2 n−1 Euler circuits, this pinpoints the exponential growth rate.

A characterization ofP4-indifference graphs

A graph is a P4-indifference graph if it admits a linear ordering ≺ on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has either a ≺ b ≺ c ≺ d or d ≺ c ≺ b ≺ a. P4-indifference graphs generalize indifference graphs and are perfectly orderable. We give a characterization of P4-indifference graphs by forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 155-162, 1999