Uniquely partitionable graphs

Jozef Bucko; Marietjie Frick; Peter Mihók; Roman Vasky

Displaying similar documents to “Uniquely partitionable graphs”

Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties

Izak Broere, Jozef Bucko, Peter Mihók (2002)

Discussiones Mathematicae Graph Theory

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Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition V₁,V₂,...,Vₙ of V(G) such that $G [V_{i}] \in_{i}$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if $_{i}$ and $_{j}$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ 1,2,...,n.

The order of uniquely partitionable graphs

Izak Broere, Marietjie Frick, Peter Mihók (1997)

Discussiones Mathematicae Graph Theory

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition V₁,...,Vₙ of V(G) such that, for each i = 1,...,n, the subgraph of G induced by $V_{i}$ has property $_{i}$ . If a graph G has a unique (₁,...,ₙ)-partition we say it is uniquely (₁,...,ₙ)-partitionable. We establish best lower bounds for the order of uniquely (₁,...,ₙ)-partitionable graphs, for various choices of ₁,...,ₙ.

Acyclic reducible bounds for outerplanar graphs

Mieczysław Borowiecki, Anna Fiedorowicz, Mariusz Hałuszczak (2009)

Discussiones Mathematicae Graph Theory

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For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. $G [V_{i}] \in_{i}$ for i = 1,...,n, 2. for every pair i,j of distinct colours the subgraph induced in G by the set of edges uv such that $u \in V_{i}$ and $v \in V_{j}$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring....

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

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We deal with the graph operator $\bar{P o w ₂}$ defined to be the complement of the square of a graph: $\bar{P o w ₂} (G) = \bar{P o w ₂ (G)}$ . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m, n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...

4-cycle properties for characterizing rectagraphs and hypercubes

Khadra Bouanane, Abdelhafid Berrachedi (2017)

Czechoslovak Mathematical Journal

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A $(0, 2)$ -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0, λ)$ -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0, 2)$ -graph. $(0, 2)$ -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0, λ)$ -graphs and more specifically...

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Jozef Bucko, Jaroslav Ivančo (1999)

Discussiones Mathematicae Graph Theory

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Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G [V_{i}]$ has the property $_{i}$ . A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a...

Clopen graphs

Stefan Geschke (2013)

Fundamenta Mathematicae

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A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their...

On the minus domination number of graphs

Hailong Liu, Liang Sun (2004)

Czechoslovak Mathematical Journal

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Let $G = (V, E)$ be a simple graph. A $3$ -valued function $f V (G) \to {- 1, 0, 1}$ is said to be a minus dominating function if for every vertex $v \in V$ , $f (N [v]) = \sum_{u \in N [v]} f (u) \geq 1$ , where $N [v]$ is the closed neighborhood of $v$ . The weight of a minus dominating function $f$ on $G$ is $f (V) = \sum_{v \in V} f (v)$ . The minus domination number of a graph $G$ , denoted by $γ^{-} (G)$ , equals the minimum weight of a minus dominating function on $G$ . In this paper, the following two results are obtained. (1) If $G$ is a bipartite graph of order $n$ , then $γ^{-} (G) \geq 4 (\sqrt{n + 1} - 1) - n .$ (2) For any negative integer $k$ and any positive integer...

The decomposability of additive hereditary properties of graphs

Izak Broere, Michael J. Dorfling (2000)

Discussiones Mathematicae Graph Theory

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If ₁,...,ₙ are properties of graphs, then a (₁,...,ₙ)-decomposition of a graph G is a partition E₁,...,Eₙ of E(G) such that $G [E_{i}]$ , the subgraph of G induced by $E_{i}$ , is in $_{i}$ , for i = 1,...,n. We define ₁ ⊕...⊕ ₙ as the property G ∈ : G has a (₁,...,ₙ)-decomposition. A property is said to be decomposable if there exist non-trivial hereditary properties ₁ and ₂ such...

Graphs with small additive stretch number

Dieter Rautenbach (2004)

Discussiones Mathematicae Graph Theory

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The additive stretch number $s_{a d d} (G)$ of a graph G is the maximum difference of the lengths of a longest induced path and a shortest induced path between two vertices of G that lie in the same component of G.We prove some properties of minimal forbidden configurations for the induced-hereditary classes of graphs G with $s_{a d d} (G) \leq k$ for some k ∈ N₀ = 0,1,2,.... Furthermore, we derive characterizations of these classes for k = 1 and k = 2.

Restrained domination in unicyclic graphs

Johannes H. Hattingh, Ernst J. Joubert, Marc Loizeaux, Andrew R. Plummer, Lucas van der Merwe (2009)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by $γ_{r} (G)$ , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then $γ_{r} (U) \geq ⎡ n / 3 ⎤$ , and provide a characterization of graphs achieving this bound.

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

Difference labelling of cacti

Martin Sonntag (2003)

Discussiones Mathematicae Graph Theory

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A graph G is a difference graph iff there exists S ⊂ IN⁺ such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = i,j:i,j ∈ V ∧ |i-j| ∈ V. It is known that trees, cycles, complete graphs, the complete bipartite graphs $K_{n, n}$ and $K_{n, n - 1}$ , pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.

Graph domination in distance two

Gábor Bacsó, Attila Tálos, Zsolt Tuza (2005)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that $D o m D o m_{u} = D o m ₂_{u}$ holds for $_{u}$ = all connected graphs without induced $P_{u}$ (u ≥ 2). (In particular,...

On generalized shift graphs

Christian Avart, Tomasz Łuczak, Vojtěch Rödl (2014)

Fundamenta Mathematicae

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In 1968 Erdős and Hajnal introduced shift graphs as graphs whose vertices are the k-element subsets of [n] = 1,...,n (or of an infinite cardinal κ ) and with two k-sets $A = a ₁, . . ., a_{k}$ and $B = b ₁, . . ., b_{k}$ joined if $a ₁ < a ₂ = b ₁ < a ₃ = b ₂ < \dots < a_{k} = b_{k - 1} < b_{k}$ . They determined the chromatic number of these graphs. In this paper we extend this definition and study the chromatic number of graphs defined similarly for other types of mutual position with respect to the underlying ordering. As a consequence of our result, we show the existence of a graph with...

Domination in partitioned graphs

Zsolt Tuza, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

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Let V₁, V₂ be a partition of the vertex set in a graph G, and let $γ_{i}$ denote the least number of vertices needed in G to dominate $V_{i}$ . We prove that γ₁+γ₂ ≤ [4/5]|V(G)| for any graph without isolated vertices or edges, and that equality occurs precisely if G consists of disjoint 5-paths and edges between their centers. We also give upper and lower bounds on γ₁+γ₂ for graphs with minimum valency δ, and conjecture that γ₁+γ₂ ≤ [4/(δ+3)]|V(G)| for δ ≤ 5. As δ gets large, however, the largest...

2-halvable complete 4-partite graphs

Dalibor Fronček (1998)

Discussiones Mathematicae Graph Theory

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A complete 4-partite graph $K_{m ₁, m ₂, m ₃, m ₄}$ is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with at most one odd part all d-halvable graphs are known. In the class of biregular graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with four odd parts (i.e., the graphs $K_{m, m, m, n}$ and $K_{m, m, n, n}$ ) all d-halvable graphs are known as well, except for the graphs $K_{m, m, n, n}$ when d = 2 and n ≠ m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with three...