arXiv subjects

Yuefang Sun

Publications and source records attributed to Yuefang Sun.

At least 37 records · Page 2Linked to original sources

arXiv2020

Semicomplete Compositions of Digraphs

Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1, \dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set $\{u_{i,j_i}\colon\, 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\colon\, u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ The composition $Q=T[H_1, \dots, H_t]$ is a semicomplete composition if $T$ is semicomplete, i.e. there is at least one arc between every pair of vertices. Digraph compositions generalize some families of digraphs, including (extended) semicomplete digraphs, quasi-transitive digraphs and lexicographic product digraphs. In particular, strong semicomplete compositions form a significant generalization of strong quasi-transitive digraphs. In this paper, we study the structural properties of semicomplete compositions and obtain results on connectivity, paths, cycles, strong spanning subdigraphs and acyclic spanning subgraphs. Our results show that this class of digraphs shares some nice properties of quasi-transitive digraphs.

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arXiv2019

Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs

A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

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arXiv2019

Extremality and Sharp Bounds for the $k$-edge-connectivity of Graphs

Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $λ_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $λ_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $λ_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will be provided such that we can get a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function $f(n,k,t)$ which is defined as the minimum size of a connected graph $G$ with order $n$ and $λ_k(G)=t$.

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arXiv2018

Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs

A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets $A_1$ and $A_2$ such that both $(V,A_1)$ and $(V,A_2)$ are strong. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ For digraph compositions $Q=T[H_1,\dots H_t]$, we obtain sufficient conditions for $Q$ to have a good decomposition and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_i$ is an arbitrary digraph with at least two vertices. For digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs $G, H$, the strong product $G\boxtimes H$ has a good decomposition.

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arXiv2018

Strong Subgraph Connectivity of Digraphs: A Survey

In this survey we overview known results on the strong subgraph $k$-connectivity and strong subgraph $k$-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and complexity, sharp bounds for strong subgraph $k$-(arc-)connectivity, minimally strong subgraph $(k, \ell)$-(arc-) connected digraphs. This survey contains several conjectures and open problems for further study.

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arXiv2018

Strong subgraph $k$-arc-connectivity

Two previous papers, arXiv:1803.00284 and arXiv:1803.00281, introduced and studied strong subgraph $k$-connectivity of digraphs obtaining characterizations, lower and upper bounds and computational complexity results for the new digraph parameter. The parameter is an analog of well-studied generalized $k$-connectivity of undirected graphs. In this paper, we introduce the concept of strong subgraph $k$-arc-connectivity of digraphs, which is an analog of generalized $k$-edge-connectivity of undirected graphs. We also obtain characterizations, lower and upper bounds and computational complexity results for this digraph parameter. Several of our results differ from those obtained for strong subgraph $k$-connectivity.

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arXiv2018

Strong subgraph $k$-connectivity bounds

Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. Strong subgraphs $D_1, \dots , D_p$ containing $S$ are said to be internally disjoint if $V(D_i)\cap V(D_j)=S$ and $A(D_i)\cap A(D_j)=\emptyset$ for all $1\le i<j\le p$. Let $κ_S(D)$ be the maximum number of internally disjoint strong digraphs containing $S$ in $D$. The strong subgraph $k$-connectivity is defined as $$κ_k(D)=\min\{κ_S(D)\mid S\subseteq V, |S|=k\}.$$ A digraph $D=(V, A)$ is called minimally strong subgraph $(k,\ell)$-connected if $κ_k(D)\geq \ell$ but for any arc $e\in A$, $κ_k(D-e)\leq \ell-1$. In this paper, we first give a sharp upper bound for the parameter $κ_k(D)$ and then study the minimally strong subgraph $(k,\ell)$-connected digraphs.

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arXiv2018

Strong Subgraph $k$-connectivity

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connectivity is, in a sense, harder to compute than generalized $k$-connectivity. However, strong subgraph $k$-connectivity can be computed in polynomial time for semicomplete digraphs and symmetric digraphs. We also provide sharp bounds on strong subgraph $k$-connectivity and pose some open questions.

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arXiv2017

An improved kernel for the cycle contraction problem

The problem of modifying a given graph to satisfy certain properties has been one of the central topics in parameterized tractability study. In this paper, we study the cycle contraction problem, which makes a graph into a cycle by edge contractions. The problem has been studied {by Belmonte et al. [IPEC 2013]} who obtained a linear kernel with at most $6k+6$ vertices. We provide an improved kernel with at most $5k+4$ vertices for it in this paper.

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arXiv2013

Characterize graphs with rainbow connection number $m-2$ and $m-3$

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of $G$, denoted by $rc(G)$, is the minimum number of colors that are needed in order to make $G$ rainbow connected. Chartrand et al. obtained that $G$ is a tree if and only if $rc(G)=m$, and it is easy to see that $G$ is not a tree if and only if $rc(G)\leq m-2$, where $m$ is the number of edge of $G$. So there is an interesting problem: Characterize the graphs $G$ with $rc(G)=m-2$. In this paper, we settle down this problem. Furthermore, we also characterize the graphs $G$ with $rc(G)=m-3$.

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arXiv2013

On the generalized (edge-)connectivity of graphs

The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized $k$-edge-connectivity $λ_k(G)$. For general $k$, the generalized $k$-edge-connectivity of a complete graph is obtained. For $k\geq 3$, tight upper and lower bounds of $κ_k(G)$ and $λ_k(G)$ are given for a connected graph $G$ of order $n$, that is, $1\leq κ_k(G)\leq n-\lceil\frac{k}{2}\rceil$ and $1\leq λ_k(G)\leq n-\lceil\frac{k}{2}\rceil$. Graphs of order $n$ such that $κ_k(G)=n-\lceil\frac{k}{2}\rceil$ and $λ_k(G)=n-\lceil\frac{k}{2}\rceil$ are characterized, respectively. Nordhaus-Gaddum-type results for the generalized $k$-connectivity are also obtained. For $k=3$, we study the relation between the edge-connectivity and the generalized 3-edge-connectivity of a graph. Upper and lower bounds of $λ_3(G)$ for a graph $G$ in terms of the edge-connectivity $λ$ of $G$ are obtained, that is, $\frac{3λ-2}{4}\leq λ_3(G)\leq λ$, and two graph classes are given showing that the upper and lower bounds are tight. From these bounds, we obtain that $λ(G)-1\leq λ_3(G)\leq λ(G)$ if $G$ is a connected planar graph, and the relation between the generalized 3-connectivity and generalized 3-edge-connectivity of a graph and its line graph.

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arXiv2012

Note on minimally $k$-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil $. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.

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arXiv2012

Graphs with large generalized 3-connectivity

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i, j$, where $1 \leq i, j \leq r$. For an integer $k$ with $2 \leq k \leq n$, the generalized $k$-connectivity $κ_k(G)$ of $G$ is the greatest positive integer $r$ such that $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $κ_2(G)$ is the connectivity of $G$. In this paper, sharp upper and lower bounds of $κ_3(G)$ are given for a connected graph $G$ of order $n$, that is, $1 \leq κ_3(G) \leq n - 2$. Graphs of order $n$ such that $κ_3(G) = n - 2, n - 3$ are characterized, respectively.

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arXiv2011

Oriented diameter and rainbow connection number of a graph

The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of $rad(G)$ and $η(G)$, where $rad(G)$ is the radius of $G$ and $η(G)$ is the smallest integer number such that every edge of $G$ is contained in a cycle of length at most $η(G)$. We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph $G$ in terms of the minimum degree of $G$.

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arXiv2011

Upper bound for the rainbow connection number of bridgeless graphs with diameter 3

A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every pair of distinct vertices of $G$ is connected by a rainbow path. It is known that for every integer $k\geq 2$ deciding if a graph $G$ has $rc(G)\leq k$ is NP-Hard, and a graph $G$ with $rc(G)\leq k$ has diameter $diam(G)\leq k$. In foregoing papers, we showed that a bridgeless graph with diameter 2 has rainbow connection number at most 5. In this paper, we prove that a bridgeless graph with diameter 3 has rainbow connection number at most 9. We also prove that for any bridgeless graph $G$ with radius $r$, if every edge of $G$ is contained in a triangle, then $rc(G)\leq 3r$. As an application, we get that for any graph $G$ with minimum degree at least 3, $rc(L(G))\leq 3 rad(L(G))\leq 3 (rad(G)+1)$.

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arXiv2011

Note on the oriented diameter of graphs with diameter 3

In 1978, Chvátal and Thomassen showed that every bridgeless graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value $f(d)$ such that every bridgeless graph $G$ with diameter $d$ has an orientation with diameter at most $f(d)$. For $d=3$, they proved that $8\leq f(d)\leq 24$. Until recently, Kwok, Liu and West improved the above bounds by proving $9\leq f(3)\leq 11$ in [P.K. Kwok, Q. Liu and D.B. West, Oriented diameter of graphs with diameter 3, J. Combin. Theory Ser.B 100(2010), 265-274]. In this paper, we determine the oriented diameter among the bridgeless graphs with diameter 3 that have minimum number of edges.

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arXiv2011

Asymptotic value of the minimal size of a graph with rainbow connection number 2

A path in an edge (vertex)-colored graph $G$, where adjacent edges (vertices) may have the same color, is called a rainbow path if no pair of edges (internal vertices) of the path are colored the same. The rainbow (vertex) connection number $rc(G)$ ($rvc(G)$) of $G$ is the minimum integer $i$ for which there exists an $i$-edge (vertex)-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. Denote by $\mathcal{G}_d(n)$ ($\mathcal{G'}_d(n)$) the set of all graphs of order $n$ with rainbow (vertex) connection number $d$, and define $e_d(n)=\min\{e(G)\,|\, G\in \mathcal{G}_d(n)\}$ ($e_d'(n)=\min\{e(G)\,|\, G\in \mathcal{G'}_d(n)\}$), where $e(G)$ denotes the number of edges in $G$. In this paper, we investigate the bounds of $e_2(n)$ and get the exact asymptotic value. i.e., $\displaystyle \lim_{n\rightarrow \infty}\frac{e_2(n)}{n\log_2 n}=1$. Meanwhile, we obtain $e'_d(n)=n-1$ for $d\geq 2$, and the equality holds if and only if $G$ is such a graph such that deleting all leaves of $G$ results in a tree of order $d$.

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arXiv2011

The generalized 3-connectivity of Cartesian product graphs

The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let $S$ be a nonempty set of vertices of $G$, a collection $\{T_1,T_2,...,T_r\}$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i,j$, where $1\leq i,j\leq r$. For an integer $k$ with $2\leq k\leq n$, the $k$-connectivity $κ_k(G)$ of $G$ is the greatest positive integer $r$ for which $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $κ_2(G)=κ(G)$ is the connectivity of $G$. Sabidussi showed that $κ(G\Box H) \geq κ(G)+κ(H)$ for any two connected graphs $G$ and $H$. In this paper, we first study the 3-connectivity of the Cartesian product of a graph $G$ and a tree $T$, and show that $(i)$ if $κ_3(G)=κ(G)\geq 1$, then $κ_3(G\Box T)\geq κ_3(G)$; $(ii)$ if $1\leq κ_3(G)< κ(G)$, then $κ_3(G\Box T)\geq κ_3(G)+1$. Furthermore, for any two connected graphs $G$ and $H$ with $κ_3(G)\geqκ_3(H)$, if $κ(G)>κ_3(G)$, then $κ_3(G\Box H)\geq κ_3(G)+κ_3(H)$; if $κ(G)=κ_3(G)$, then $κ_3(G\Box H)\geq κ_3(G)+κ_3(H)-1$. Our result could be seen as a generalization of Sabidussi's result. Moreover, all the bounds are sharp.

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