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A characterization of the set of all shortest paths in a connected graph

Ladislav Nebeský (1994)

Mathematica Bohemica

Let G be a (finite undirected) connected graph (with no loop or multiple edge). The set of all shortest paths in G is defined as the set of all paths ξ , then the lenght of ξ does not exceed the length of ς . While the definition of is based on determining the length of a path. Theorem 1 gives - metaphorically speaking - an “almost non-metric” characterization of : a characterization in which the length of a path greater than one is not considered. Two other theorems are derived from Theorem...

A conjecture on cycle-pancyclism in tournaments

Hortensia Galeana-Sánchez, Sergio Rajsbaum (1998)

Discussiones Mathematicae Graph Theory

Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Cₖ of length k in T we denote I γ ( C ) = | A ( γ ) A ( C ) | , the number of arcs that γ and Cₖ have in common. Let f ( k , T , γ ) = m a x I γ ( C ) | C T and f(n,k) = minf(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T. In previous papers we gave...

A decomposition of gallai multigraphs

Alexander Halperin, Colton Magnant, Kyle Pula (2014)

Discussiones Mathematicae Graph Theory

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees...

A Fan-Type Heavy Pair Of Subgraphs For Pancyclicity Of 2-Connected Graphs

Wojciech Wideł (2016)

Discussiones Mathematicae Graph Theory

Let G be a graph on n vertices and let H be a given graph. We say that G is pancyclic, if it contains cycles of all lengths from 3 up to n, and that it is H-f1-heavy, if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies [...] min⁡dG(u),dG(v)≥n+12 min { d G ( u ) , d G ( v ) } n + 1 2 . In this paper we prove that every 2-connected K1,3, P5-f1-heavy graph is pancyclic. This result completes the answer to the problem of finding f1-heavy pairs of subgraphs implying pancyclicity...

A linear algorithm for the two paths problem on permutation graphs

C.P. Gopalakrishnan, C. Pandu Rangan (1995)

Discussiones Mathematicae Graph Theory

The 'two paths problem' is stated as follows. Given an undirected graph G = (V,E) and vertices s₁,t₁;s₂,t₂, the problem is to determine whether or not G admits two vertex-disjoint paths P₁ and P₂ connecting s₁ with t₁ and s₂ with t₂ respectively. In this paper we give a linear (O(|V|+ |E|)) algorithm to solve the above problem on a permutation graph.

A Linear Time Algorithm for Computing Longest Paths in Cactus Graphs

Markov, Minko, Ionut Andreica, Mugurel, Manev, Krassimir, Tapus, Nicolae (2012)

Serdica Journal of Computing

ACM Computing Classification System (1998): G.2.2.We propose an algorithm that computes the length of a longest path in a cactus graph. Our algorithm can easily be modified to output a longest path as well or to solve the problem on cacti with edge or vertex weights. The algorithm works on rooted cacti and assigns to each vertex a two-number label, the first number being the desired parameter of the subcactus rooted at that vertex. The algorithm applies the divide-and-conquer approach and computes...

A metric for graphs

Vladimír Baláž, Jaroslav Koča, Vladimír Kvasnička, Milan Sekanina (1986)

Časopis pro pěstování matematiky

A modification of the median of a tree

Bohdan Zelinka (1993)

Mathematica Bohemica

The concept of median of a tree is modified, considering only distances from the terminal vertices instead of distances from all vertices.

A new approach to chordal graphs

Ladislav Nebeský (2007)

Czechoslovak Mathematical Journal

By a chordal graph is meant a graph with no induced cycle of length 4 . By a ternary system is meant an ordered pair ( W , T ) , where W is a finite nonempty set, and T W × W × W . Ternary systems satisfying certain axioms (A1)–(A5) are studied in this paper; note that these axioms can be formulated in a language of the first-order logic. For every finite nonempty set W , a bijective mapping from the set of all connected chordal graphs G with V ( G ) = W onto the set of all ternary systems ( W , T ) satisfying the axioms (A1)–(A5) is...

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