Balanced problems on graphs with categorization of edges

Štefan Berežný; Vladimír Lacko

Displaying similar documents to “Balanced problems on graphs with categorization of edges”

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...

New edge neighborhood graphs

Ali A. Ali, Salar Y. Alsardary (1997)

Czechoslovak Mathematical Journal

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Let $G$ be an undirected simple connected graph, and $e = u v$ be an edge of $G$ . Let $N_{G} (e)$ be the subgraph of $G$ induced by the set of all vertices of $G$ which are not incident to $e$ but are adjacent to $u$ or $v$ . Let $𝒩_{e}$ be the class of all graphs $H$ such that, for some graph $G$ , $N_{G} (e) ≅ H$ for every edge $e$ of $G$ . Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in $𝒩_{e}$ . Balasubramanian and Alsardary [1] obtained some other graphs in $𝒩_{e}$ . In this paper we given some new graphs in $𝒩_{e}$ .

On perfect and unique maximum independent sets in graphs

Lutz Volkmann (2004)

Mathematica Bohemica

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A perfect independent set $I$ of a graph $G$ is defined to be an independent set with the property that any vertex not in $I$ has at least two neighbors in $I$ . For a nonnegative integer $k$ , a subset $I$ of the vertex set $V (G)$ of a graph $G$ is said to be $k$ -independent, if $I$ is independent and every independent subset $I^{'}$ of $G$ with $| I^{'} | \geq | I | - (k - 1)$ is a subset of $I$ . A set $I$ of vertices of $G$ is a super $k$ -independent set of $G$ if $I$ is $k$ -independent in the graph $G [I, V (G) - I]$ , where $G [I, V (G) - I]$ is the bipartite graph obtained from $G$ by deleting...

Graceful signed graphs

Mukti Acharya, Tarkeshwar Singh (2004)

Czechoslovak Mathematical Journal

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A $(p, q)$ -sigraph $S$ is an ordered pair $(G, s)$ where $G = (V, E)$ is a $(p, q)$ -graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^{+}$ and $E^{-}$ consist of $m$ positive and $n$ negative edges of $G$ , respectively, where $m + n = q$ . Given positive integers $k$ and $d$ , $S$ is said to be $(k, d)$ -graceful if the vertices of $G$ can be labeled with distinct integers from the set ${0, 1, \dots, k + (q - 1) d}$ such that when each edge $u v$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to...

Decomposition of complete graphs into $(0, 2)$ -prisms

Sylwia Cichacz, Soleh Dib, Dalibor Fronček (2014)

Czechoslovak Mathematical Journal

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R. Frucht and J. Gallian (1988) proved that bipartite prisms of order $2 n$ have an $α$ -labeling, thus they decompose the complete graph $K_{6 n x + 1}$ for any positive integer $x$ . We use a technique called the $ρ^{+}$ -labeling introduced by S. I. El-Zanati, C. Vanden Eynden, and N. Punnim (2001) to show that also some other families of 3-regular bipartite graphs of order $2 n$ called generalized prisms decompose the complete graph $K_{6 n x + 1}$ for any positive integer $x$ .

Destroying symmetry by orienting edges: complete graphs and complete bigraphs

Frank Harary, Michael S. Jacobson (2001)

Discussiones Mathematicae Graph Theory

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Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group. We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs $K_{s, t}$ , s ≤ t, this number does not always exist. We determine for s ≤ 4 the values of t for which this number does exist.

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.

Labeling the vertex amalgamation of graphs

Ramon M. Figueroa-Centeno, Rikio Ichishima, Francesc A. Muntaner-Batle (2003)

Discussiones Mathematicae Graph Theory

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A graph G of size q is graceful if there exists an injective function f:V(G)→ 0,1,...,q such that each edge uv of G is labeled |f(u)-f(v)| and the resulting edge labels are distinct. Also, a (p,q) graph G with q ≥ p is harmonious if there exists an injective function $f : V (G) \to Z_{q}$ such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct, whereas G is felicitous if there exists an injective function $f : V (G) \to Z_{q + 1}$ such that each edge uv of G is labeled f(u) + f(v) mod...

The signed matchings in graphs

Changping Wang (2008)

Discussiones Mathematicae Graph Theory

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Let G be a graph with vertex set V(G) and edge set E(G). A signed matching is a function x: E(G) → -1,1 satisfying $\sum_{e \in E_{G} (v)} x (e) \leq 1$ for every v ∈ V(G), where $E_{G} (v) = u v \in E (G) | u \in V (G)$ . The maximum of the values of $\sum_{e \in E (G)} x (e)$ , taken over all signed matchings x, is called the signed matching number and is denoted by β’₁(G). In this paper, we study the complexity of the maximum signed matching problem. We show that a maximum signed matching can be found in strongly polynomial-time. We present sharp upper and lower bounds on β’₁(G) for...

Independent cycles and paths in bipartite balanced graphs

Beata Orchel, A. Paweł Wojda (2008)

Discussiones Mathematicae Graph Theory

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Bipartite graphs G = (L,R;E) and H = (L’,R’;E’) are bi-placeabe if there is a bijection f:L∪R→ L’∪R’ such that f(L) = L’ and f(u)f(v) ∉ E’ for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy ||G|| ≤ 2p-3 and ||H|| ≤ 2p-2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that...

Histories in path graphs

Ludovít Niepel (2007)

Discussiones Mathematicae Graph Theory

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For a given graph G and a positive integer r the r-path graph, $P_{r} (G)$ , has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P_{r}^{k} (G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_{r}^{k} (G)$ . The k-history $P_{r}^{- k} (H)$ is a subgraph of G that is induced by all edges that take part in the recursive...