A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. We consider Balanced Connected Partitions (BCP), where the two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have $K_{1,c}$ as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, due to the (3-)claw-freeness of line graphs, this also implies a 2-approximations for the edge-partition version of BCP in general graphs.
In the 1970s Győri and Lovász showed for natural numbers $w_1,\dots,w_k$ where $\sum_i w_i$ is the vertex size, that if $G$ is k-connected, then there exist a connected k-partition with part sizes $w_1,\dots,w_k$. However, to this day no polynomial algorithm to compute such partitions exists for k>4. Towards finding such a partition $T_1,\dots, T_k$, we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each $w_i$ is greater than the weight of the heaviest vertex. In particular, we give a 3-approximation for both the lower and the upper bounded version i.e. we guarantee that each $T_i$ has weight at least $\frac{w_i}{3}$ or that each $T_i$ has weight most $3w_i$, respectively. Also, we present a both-side bounded version that produces a connected partition where each $T_i$ has size at least $\frac{w_i}{3}$ and at most $\max(\{r,3\}) w_i$, where $r \geq 1$ is the ratio between the largest and smallest value in $w_1, \dots, w_k$. In particular for the balanced version, i.e.~$w_1=w_2=, \dots,=w_k$, this gives a partition with $\frac{1}{3}w_i \leq w(T_i) \leq 3w_i$.
The claw number of a graph $G$ is the largest number $v$ such that $K_{1,v}$ is an induced subgraph of $G$. Interval graphs with claw number at most $v$ are cluster graphs when $v = 1$, and are proper interval graphs when $v = 2$. Let $κ(n,v)$ be the smallest number $k$ such that every interval graph with $n$ vertices admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. Let $\checkκ(w,v)$ be the smallest number $k$ such that every interval graph with claw number $w$ admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. We show that $κ(…
In 2006 Bar{á}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with edge-connectivity $4$ which the smallest one contains $24$ vertices. In this note, we first give a smaller counterexample having only $18$ vertices and next construct a family of planar $4$-connected essentially $6$-edge-connected $4$-regular simple graphs of size divisible by three with no claw-decompositions. This result provides the sharpness for two known …
Broersma and Veldman proved that every 2-connected claw-free and $P_6$-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and $P_6$-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and $P_6$-\emph{o}-heavy but not hamiltonian. In this paper we further give some Ore-type degree conditions restricting to induced $P_6$'s of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.
A graph $G$ is said to be $k$-$γ$-vertex critical if the domination numbers $γ(G)$ of $G$ is $k$ and $γ(G - v) < k$ for any vertex $v$ of $G$. Similarly, A graph $G$ is said to be $k$-$γ_{c}$-vertex critical if the connected domination numbers $γ_{c}(G)$ of $G$ is $k$ and $γ_{c}(G - v) < k$ for any vertex $v$ of $G$. The problem of interest is to determine whether or not $2$-connected $k$-$γ$-vertex critical graphs are Hamiltonian. In this paper, for all $k \geq 3$, we provide a $2$-connected $k$-$γ$-vertex critical graph which is non-Hamiltonian. We prove that every $2$-connected $3$-$γ$-vert…
An ordering of the vertices of a graph is \emph{connected} if every vertex (but the first) has a neighbor among its predecessors. The greedy colouring algorithm of a graph with a connected order consists in taking the vertices in order, and assigning to each vertex the smallest available colour. A graph is \emph{good} if the greedy algorithm on every connected order gives every connected induced subgraph of it an optimal colouring. We give the characterization of good claw-free graphs in terms of minimal forbidden induced subgraphs.
Let $G$ be a connected claw-free graph on $n$ vertices and $\overline{G}$ be its complement graph. Let $μ(G)$ be the spectral radius of $G$. Denote by $N_{n-3,3}$ the graph consisting of $K_{n-3}$ and three disjoint pendent edges. In this note we prove that: (1) If $μ(G)\geq n-4$, then $G$ is traceable unless $G=N_{n-3,3}$. (2) If $μ(\overline{G})\leq μ(\overline{N_{n-3,3}})$ and $n\geq 24$, then $G$ is traceable unless $G=N_{n-3,3}$. Our works are counterparts on claw-free graphs of previous theorems due to Lu et al., and Fiedler and Nikiforov, respectively.
In 1962, Erdős proved that if a graph $G$ with $n$ vertices satisfies $$ e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\}, $$ where the minimum degree $δ(G)\geq k$ and $1\leq k\leq(n-1)/2$, then it is Hamiltonian. For $n \geq 2k+1$, let $E^k_n=K_{k}\vee (kK_1+K_{n-2k})$, where "$\vee$" is the "join" operation. One can observe $e(E^k_n)=\binom{n-k}{2}+k^2$ and $E^k_n$ is not Hamiltonian. As $E^k_n$ contains induced claws for $k\geq 2$, a natural question is to characterize all 2-connected claw-free non-Hamiltonian graphs with th…
A cycle $C$ of length $k$ in graph $G$ is extendable if there is another cycle $C'$ in $G$ with $V(C) \subset V(C')$ and length $k+1$. A graph is cycle extendable if every non-Hamiltonian cycle is extendable. In 1990 Hendry conjectured that any Hamiltonian chordal graph (a Hamiltonian graph with no induced cycle of length greater than three) is cycle extendable, and this conjecture has been verified for Hamiltonian chordal graphs which are interval graphs, planar graphs, and split graphs. We prove that any 2-connected claw-free chordal graph is cycle extendable.
An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with given two different vertices s, t and an edge e to have (1)a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian s- and st-paths containing edge e when G has connectivity one, and (4) a Hamiltonian cycle containing e when G is 2-connected. These results imply that a connected {claw, net}-free graph has a Hamiltonian path and a 2-connected {claw, net}-free graph has a Hamiltonian cycle…
Hoffmann-Ostenhof's Conjecture states that states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a $2$-regular subgraph. In this paper, we show that the conjecture holds for claw-free cubic graphs.
The connected graph of degree sequence 3,3,3,1,1,1 is called a net, and the vertices of degree 1 in a net is called its endvertices. Broersma conjectured in 1993 that a 2-connected graph G with no induced K_{1,3} is hamiltonian if every endvertex of each induced net of G has degree at least (|V(G)|-2)/3. In this paper we prove this conjecture in the affirmative.
Let $T$ be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of $T$ is denoted by $Leaf(T)$. The subtree $T-Leaf(T)$ of $T$ is called the stem of $T$ and denoted by $Stem(T).$ In this paper, we give two sufficient conditions for a connected claw-free graph to have a spanning tree whose stem has a bounded number of branch vertices, and those conditions are best possible. As corollaries of main results we also give some conditions to show that a connected claw-free graph has a spanning tree whose stem is a s…
Hoffmann-Ostenhof's Conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a $2$-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and $4$-chordal subcubic graphs.
In this article, we will prove that if $G$ is a connected claw-free graph and either $σ_6(G)\geq |G|-5$ or $σ_7(G)\geq |G|-2$, here $σ_k(G)$ is the minimmum degree sum of $k$ independent vertices in $G$, then $G$ has a spanning tree with at most two branch vertices.
In this paper, we first show that the power domination number of a connected $4$-regular claw-free graph on $n$ vertices is at most $\frac{n+1}{5}$, and the bound is sharp. The statement partly disprove the conjecture presented by Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a dynamic programming style linear-time algorithm for weighted power domination problem in trees.
An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) denote the number of vertices of G. A graph is called claw-free if it does not have a subgraph isomorphic to the graph with 4 vertices and 3 edges having a common vertex. Our results include the following. Let G$ be a 3-connected claw-free graph, x be a vertex, e = xy be an edge, and P be a 3-vertex path in G. Then (c1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (c2) if v(G) = 1 mod 3, then G - x has a L-factor, (c3) if v(G) = 2 mod 3, then G - x -y has an L-factor, (…
Let $G$ be a graph on $n\geq 3$ vertices. A graph $G$ is almost distance-hereditary if each connected induced subgraph $H$ of $G$ has the property $d_{H}(x,y)\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\in V(H)$. A graph $G$ is called 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of $G$ isomorphic to $K_{1,3}$ (a claw) has (have) degree at least $n/2$, and called claw-heavy if each claw of $G$ has a pair of end vertices with degree sum at least $n$. Thus every 2-heavy graph is claw-heavy. In this paper we prove the following two results: (1) Every 2-c…
A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs.
In this paper, we give a series of couterexamples to negate a conjecture and hence answer an open question on the $k$-power domination of regular graphs (see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]). Furthermore, we focus on the study of $k$-power domination of claw-free graphs. We show that for $l\in\{2,3\}$ and $k\ge l$, the $k$-power domination number of a connected claw-free $(k+l+1)$-regular graph on $n$ vertices is at most $\frac{n}{k+l+2}$, and this bound is tight.
In this research, we determine the structure of (claw, bull)-free graphs. We show that every connected (claw, bull)-free graph is either an expansion of a path, an expansion of a cycle, or the complement of a triangle-free graph; where an expansion of a graph $G$ is obtained by replacing its vertices with disjoint cliques and adding all edges between cliques corresponding to adjacent vertices of $G$. This result also reveals facts about the structure of triangle-free graphs, which might be of independent interest.
We study the class of independence complexes of claw-free graphs. The main theorem give good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is (cn/d+epsilon)-connected, where c=2/3. This can be compared with the result of Szabo and Tardos that c=1/2 is optimal with no restrictions on the graphs. Secondly, we calculate the connectivity of a family of complexes used in Babson and Kozlov's proof of Lovasz co…
In 1968, Erdös and Lovász conjectured that for every graph $G$ and all integers $s,t\geq 2$ such that $s+t-1=χ(G) > ω(G)$, there exists a partition $(S,T)$ of the vertex set of $G$ such that $χ(G|S)\geq s$ and $χ(G|T)\geq t$. For general graphs, the only settled cases of the conjecture are when $s$ and $t$ are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2 \cite{quasi-line}, line graphs \cite{line} and quasi-line graphs \cite{quasi-line}. In this paper, we consider the conjecture for claw-free graphs and present some progress on i…
A graph is called \emph{claw-free} if it contains no induced subgraph isomorphic to $K_{1,3}$. Matthews and Sumner proved that a 2-connected claw-free graph $G$ is hamiltonian if every vertex of it has degree at least $(|V(G)|-2)/3$. At the workshop C\&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of $N$ (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of ind…
The zero forcing number of a simple graph, written $Z(G)$, is a NP-hard graph invariant which is the result of the zero forcing color change rule. This graph invariant has been heavily studied by linear algebraists, physicists, and graph theorist. It's broad applicability and interesting combinatorial properties have attracted the attention of many researchers. Of particular interest, is that of bounding the zero forcing number from above. In this paper we show a surprising relation between the zero forcing number of a graph and the independence number of a graph, denoted $α(G)$. Our main theo…
We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly observable condition on their partitioned genus distributions, even though it had been proved previously that iterative amalgamation does not always preserve real-rootedness of the genus polynomial of the i…
A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set $S$ is called a forcing set (zero forcing set) of $G$ if, by iteratively applying the forcing process, every vertex in $G$ becomes colored. If the initial set $S$ has the added property that it induces a subgraph of $G$ without isolated vertices, then $S$ is called a total forcing set in $G$. The…
Let $G$ be a graph and $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is an edge coloring such that each color appears at each vertex $v\in V(G)$ at most $f (v)$ times. The minimum number of colors needed to $f$-color $G$ is called the $f$-chromatic index of $G$ and is denoted by $χ'_{f}(G)$. It was shown that for every graph $G$, $Δ_{f}(G)\le χ'_{f}(G)\le Δ_{f}(G)+1$, where $Δ_{f}(G)=\max_{v\in V(G)} \lceil \frac{d_G(v)}{f(v)} \rceil$. A graph $G$ is said to be $f$-Class $1$ if $χ'_{f}(G)=Δ_{f}(G)$, and $f$-Class $2$, otherwise. Also, $G_{Δ_f}$ is the induced sub…
In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains a claw ($K_{1,3}$) or a net (a fixed six-vertex graph) as an induced subgraph. This implies that subject to being connected, these two are the only minimal (under taking induced subgraphs) graphs with no Hamiltonian path. Brousek (1998) characterized the minimal graphs that are $2$-connected, non-Hamiltonian and do not contain the claw as an induced subgraph. We characterize the minimal graphs that are $2$-connected and non-Hamiltonian for two classes of graphs: (1) split graphs, (…
Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and $S$ can be extended to a $k$-link between $v$ and $S$ such that the endvertices of the $(k-1)$-link are also the endvertices of the $k$-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets $S_1$ and $S_2$ are $k$-linked, then a $t$-link ($t< k$) between two disjoint sets $S_1$ and $S_2$ can be extended to a $k$-l…
We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph ${\mathcal H} ^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hy…
We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.
A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness.
We describe ${\rm Forb}\{K_{1,3}, \bar {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $\bar{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths, and of the complements of these graphs. Considering the hypergraph ${\mathcal H}^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the r…
A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms are known for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to t…
Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of $G$ has degree at least $n/2$. In this note, we show that every 2-connected $o$-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et …
Yang et al. proved that every 3-connected, essentially 11-connected line graph is Hamilton-connected. This was extended by Li and Yang to 3-connected, essentially 10-connected graphs. Strengthening their result further, we prove that 3-connected, essentially 9-connected line graphs are Hamilton-connected. We use a method based on quasigraphs in combination with the discharging technique. The result extends to claw-free graphs.
An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3…
Let $G$ be a graph on $n\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced claw of $G$ has degree at least $n/2$, and claw-\emph{o}-heavy if each induced claw of it has a pair of end vertices with degree sum at least $n$. In this paper we prove two results: (1) Every 2-connected claw-$o$-heavy graph $G$ is Hamiltonian if every pair of vertices $u,v$ in a subgraph $H\cong Z_1$ contained in an induced subgraph $Z_2$ of $G$ with $d_{…
For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$ is an induced subgraph of some of $P_6$, $K_1 \cup P_5$, $2P_3$, $Z_2$ or $K_1 \cup Z_1$. Furthermore, for $X$ chosen as $2K_1 \cup K_3$, we list all eight imperfect graphs belonging to the class; and for every other choice of $X$, we show that there are infinitely many such graphs. In addition, for $X$ chosen as $B_{1,2}$, we describe the structure of all …
We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph $G$ is $H$-free for some graph $H$ if $G$ contains no induced subgraph isomorphic to $H$. It is known that Connected Vertex Cover is NP-complete even for $H$-free graphs if $H$ contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if $H$ contains a cycle or claw. In the…
Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw…
A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erd\H os and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph $G$ with chromatic number $t$ is double-critical, then $G$ is the complete graph on $t$ vertices. This has been verified for $t \le 5$, but remains open for $t \ge 6$. In this paper…
