Displaying similar documents to “On the null space of a Colin de Verdière matrix”

The vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), I D [ S ] = x , y S I D [ x , y ] . A set S of vertices is a detour convex set if I D [ S ] = S . The detour convex hull [ S ] D is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

The extremal irregularity of connected graphs with given number of pendant vertices

Xiaoqian Liu, Xiaodan Chen, Junli Hu, Qiuyun Zhu (2022)

Czechoslovak Mathematical Journal

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The irregularity of a graph G = ( V , E ) is defined as the sum of imbalances | d u - d v | over all edges u v E , where d u denotes the degree of the vertex u in G . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with n vertices and p pendant vertices ( 1 p n - 1 ), and characterize the corresponding extremal graphs.

On the order of certain close to regular graphs without a matching of given size

Sabine Klinkenberg, Lutz Volkmann (2007)

Czechoslovak Mathematical Journal

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A graph G is a { d , d + k } -graph, if one vertex has degree d + k and the remaining vertices of G have degree d . In the special case of k = 0 , the graph G is d -regular. Let k , p 0 and d , n 1 be integers such that n and p are of the same parity. If G is a connected { d , d + k } -graph of order n without a matching M of size 2 | M | = n - p , then we show in this paper the following: If d = 2 , then k 2 ( p + 2 ) and (i) n k + p + 6 . If d 3 is odd and t an integer with 1 t p + 2 , then (ii) n d + k + 1 for k d ( p + 2 ) , (iii) n d ( p + 3 ) + 2 t + 1 for d ( p + 2 - t ) + t k d ( p + 3 - t ) + t - 3 , (iv) n d ( p + 3 ) + 2 p + 7 for k p . If d 4 is even, then (v) n d + k + 2 - η for k d ( p + 3 ) + p + 4 + η , (vi) n d + k + p + 2 - 2 t = d ( p + 4 ) + p + 6 for k = d ( p + 3 ) + 4 + 2 t and p 1 ,...

Stronger bounds for generalized degrees and Menger path systems

R.J. Faudree, Zs. Tuza (1995)

Discussiones Mathematicae Graph Theory

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For positive integers d and m, let P d , m ( G ) denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that P d , m ( G ) is satisfied have been investigated. In particular, it has been shown,...

Domination Subdivision Numbers

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, David P. Jacobs, James Knisely, Lucas C. van der Merwe (2001)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number s d γ ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that 1 s d γ ( G ) 3 for any graph G. We give a counterexample to this conjecture. On the other hand,...

The geodetic number of strong product graphs

A.P. Santhakumaran, S.V. Ullas Chandran (2010)

Discussiones Mathematicae Graph Theory

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For two vertices u and v of a connected graph G, the set I G [ u , v ] consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets I G [ u , v ] for u,v ∈ S is denoted by I G [ S ] . A set S ⊆ V(G) is a geodetic set if I G [ S ] = V ( G ) and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.

Proper connection number of bipartite graphs

Jun Yue, Meiqin Wei, Yan Zhao (2018)

Czechoslovak Mathematical Journal

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An edge-colored graph G is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph G , denoted by pc ( G ) , is the smallest number of colors that are needed to color the edges of G in order to make it proper connected. In this paper, we obtain the sharp upper bound for pc ( G ) of a general bipartite graph G and a series of extremal graphs. Additionally, we give a proper 2 -coloring for a connected bipartite graph G having δ ( G ) 2 and a dominating...

On-line ranking number for cycles and paths

Erik Bruoth, Mirko Horňák (1999)

Discussiones Mathematicae Graph Theory

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A k-ranking of a graph G is a colouring φ:V(G) → 1,...,k such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number χ * r ( G ) of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that χ * r ( P ) < 3 l o g n for n ≥ 2....

Edge-sum distinguishing labeling

Jan Bok, Nikola Jedličková (2021)

Commentationes Mathematicae Universitatis Carolinae

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We study edge-sum distinguishing labeling, a type of labeling recently introduced by Z. Tuza (2017) in context of labeling games. An ESD labeling of an n -vertex graph G is an injective mapping of integers 1 to l to its vertices such that for every edge, the sum of the integers on its endpoints is unique. If l equals to n , we speak about a canonical ESD labeling. We focus primarily on structural properties of this labeling and show for several classes of graphs if they have or do not...

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

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The domination subdivision number s d γ ( G ) of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

Domination numbers in graphs with removed edge or set of edges

Magdalena Lemańska (2005)

Discussiones Mathematicae Graph Theory

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It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number γ w and the connected domination number γ c , i.e., we show that γ w ( G ) γ w ( G - e ) γ w ( G ) + 1 and γ c ( G ) γ c ( G - e ) γ c ( G ) + 2 if G and G-e are connected. Additionally we show that γ w ( G ) γ w ( G - E ) γ w ( G ) + p - 1 and γ c ( G ) γ c ( G - E ) γ c ( G ) + 2 p - 2 if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order...

Full domination in graphs

Robert C. Brigham, Gary Chartrand, Ronald D. Dutton, Ping Zhang (2001)

Discussiones Mathematicae Graph Theory

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For each vertex v in a graph G, let there be associated a subgraph H v of G. The vertex v is said to dominate H v as well as dominate each vertex and edge of H v . A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number γ F H ( G ) . A full dominating set of G of cardinality γ F H ( G ) is called a γ F H -set of G. We study three types of full domination in...

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

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The line graph of a graph G , denoted by L ( G ) , has E ( G ) as its vertex set, where two vertices in L ( G ) are adjacent if and only if the corresponding edges in G have a vertex in common. For a graph H , define σ ¯ 2 ( H ) = min { d ( u ) + d ( v ) : u v E ( H ) } . Let H be a 2-connected claw-free simple graph of order n with δ ( H ) 3 . We show that, if σ ¯ 2 ( H ) 1 7 ( 2 n - 5 ) and n is sufficiently large, then either H is traceable or the Ryjáček’s closure cl ( H ) = L ( G ) , where G is an essentially 2 -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

Turán number of two vertex-disjoint copies of cliques

Caiyun Hu (2024)

Czechoslovak Mathematical Journal

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The Turán number of a given graph H , denoted by ex ( n , H ) , is the maximum number of edges in an H -free graph on n vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number ex ( n , K p K q ) of a vertex-disjoint union of cliques K p and K q for all values of n .