arXiv subjects
Publications and source records attributed to Sandi Klavžar.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$ we have $D'(G) \leq D(G) + 1$. In this note we characterize trees for which this inequality is sharp. We also show that if $G$ is a connected unicyclic graph, then $D'(G) = D(G)$.
The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\rm sg}(G \,\square\, H)$ is determined, as well as exact values for $K_m \,\square\, K_n$, $K_{1, k} \,\square\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in [k]$, where each $V_i$ is an $i$-packing. In this paper, we investigate for a given triple $(a,b,c)$ of positive integers whether there exists a graph $G$ such that $ω(G) = a$, $χ(G) = b$, and $χ_ρ(G) = c$. If so, we say that $(a, b, c)$ is realizable. It is proved that $b=c\ge 3$ implies $a=b$, and that triples $(2,k,k+1)$ and $(2,k,k+2)$ are not realizable as soon as $k\ge 4$. Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on $χ_ρ(G)$ in terms of $Δ(G)$ and $α(G)$ is also proved.
Let X $\subseteq$ {0, 1} n. Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x $\in$ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial D G,u (x, y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Q k at a given distance from the vertex u. It is proved that if G is a daisy cube, then D G,0 n (x, y) = C G (x + y -- 1), where C G (x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then D G,u (x, --x) = 1 holds for every vertex u in G.
In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. The length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities.
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $Π_1,\ldots,Π_k$, where $Π_i$, $i\in [k]$, is an $i$-packing. The following conjecture is posed and studied: if $G$ is a subcubic graph, then $χ_ρ(S(G))\le 5$, where $S(G)$ is the subdivision of $G$. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if $G$ is a generalized prism of a cycle, then $G$ is $(1,1,2,2)$-colorable if and only if $G$ is not the Petersen graph. The validity of the conjecture is further proved for graphs that can be obtained from generalized prisms in such a way that one of the two $n$-cycles in the edge set of a generalized prism is replaced by a union of cycles among which at most one is a 5-cycle. The packing chromatic number of graphs obtained by subdividing each of its edges a fixed number of times is also considered.
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. It is proved that in the class of subcubic graphs the packing chromatic number is bigger than $13$, thus answering an open problem from [Gastineau, Togni, $S$-packing colorings of cubic graphs, Discrete Math.\ 339 (2016) 2461--2470]. In addition, the packing chromatic number is investigated with respect to several local operations. In particular, if $S_e(G)$ is the graph obtained from a graph $G$ by subdividing its edge $e$, then $\left\lfloor χ_ρ(G)/2 \right\rfloor +1 \le χ_ρ(S_e(G)) \le χ_ρ(G)+1$.
A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.
With the aid of hypergraph transversals it is proved that $γ_t(Q_{n+1}) = 2γ(Q_n)$, where $γ_t(G)$ and $γ(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the $n$-dimensional hypercube. More generally, it is shown that if $G$ is a bipartite graph, then $γ_t(G \square K_2) = 2γ(G)$. Further, we show that the bipartite condition is essential by constructing, for any $k \ge 1$, a (non-bipartite) graph $G$ such that $γ_t (G \square K_2 ) = 2γ(G) - k$. Along the way several domination-type identities for hypercubes are also obtained.
The game domination number is a graph invariant that arises from a game, which is related to graph domination in a similar way as the game chromatic number is related to graph coloring. In this paper we show that verifying whether the game domination number of a graph is bounded by a given integer is PSPACE-complete. This contrasts the situation of the game coloring problem whose complexity is still unknown.
The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to $D_k(G)$ for some graph $G$ and some positive integer $k$. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if $G$ is such a graph of order $n\ge 2$ and with $G\cong D_k(G)$, then $k=2$ and $G=K_{1,n-1}$ for some $n\ge 4$. It is also proved that for a given $r$ there exist only a finite number of $r$-regular, connected dominating graphs of connected graphs. In particular, $C_6$ and $C_8$ are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.
An algorithmic upper bound on the domination number $γ$ of graphs in terms of the order $n$ and the minimum degree $δ$ is proved. It is demonstrated that the bound improves best previous bounds for any $5\le δ\le 50$. In particular, for $δ=5$, Xing et al.\ proved in 2006 that $γ\le 5n/14 < 0.3572 n$. This bound is improved to $0.3440 n$. For $δ=6$, Clark et al.\ in 1998 established $γ<0.3377 n$, while Biró et al. recently improved it to $γ<0.3340 n$. Here the bound is further improved to $γ< 0.3159 n$. For $δ=7$, the best earlier bound $0.3 088 n$ is improved to $γ< 0.2927 n$.
The Fibonacci cube $Γ_n$ is obtained from the $n$-cube $Q_n$ by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube $Λ_n$ is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of the sizes of the vertex orbits of $Λ_n$ is $\{k \ge 1;\ k \divides n\} \cup\, \{k \ge 18;\ k \divides 2n\}$, the number of the vertex orbits of $Λ_n$ of size $k$, where $k$ is odd and divides $n$, is equal to $\sum_{d\divides k}μ\left(\frac{k}{d}\right) F_{\lfloor \frac{d}{2}\rfloor + 2}$, and the number of the edge orbits of $Λ_n$ is equal to the number of the vertex orbits of $Γ_{n-3}$. Dihedral transformations of strings and primitive strings are essential tools to prove these results.
Let $G$ be a graph and let $m_{ij}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The {\em $M$-polynomial} of $G$ is introduced with $\displaystyle{M(G;x,y) = \sum_{i\le j} m_{ij}(G)x^iy^j}$. It is shown that degree-based topological indices can be routinely computed from the polynomial, thus reducing the problem of their determination in each particular case to the single problem of determining the $M$-polynomial. The new approach is also illustrated with examples.
The generalized Fibonacci cube $Q_d(f)$ is the subgraph of the $d$-cube $Q_d$ induced on the set of all strings of length $d$ that do not contain $f$ as a substring. It is proved that if $Q_d(f) \cong Q_d(f')$ then $|f|=|f'|$. The key tool to prove this result is a result of Guibas and Odlyzko about the autocorrelation polynomial associated to a binary string. It is also proved that there exist pairs of strings $f, f'$ such that $Q_d(f) \cong Q_d(f')$, where $|f| \ge \frac{2}{3}(d+1)$ and $f'$ cannot be obtained from $f$ by its reversal or binary complementation. Strings $f$ and $f'$ with $|f|=|f'|=d-1$ for which $Q_d(f) \cong Q_d(f')$ are characterized.
The domination game is played on a graph $G$ by two players, named Dominator and Staller. They alternatively select vertices of $G$ such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator's goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Game 2 are variants of the game in which Dominator and Staller has the first move, respectively. The game domination number $γ_g(G)$, and the Staller-start game domination number $γ_g'(G)$, is the number of vertices chosen in Game 1 and Game 2, respectively. It is proved that if $e\in E(G)$, then $|γ_g(G) - γ_g(G-e)| \le 2$ and $|γ_g'(G) - γ_g'(G-e)| \le 2$, and that each of the possibilities here is realizable by connected graphs $G$ for all values of $γ_g(G)$ and $γ_g'(G)$ larger than 5. For the remaining small values it is either proved that realizations are not possible or realizing examples are provided. It is also proved that if $v\in V(G)$, then $γ_g(G) - γ_g(G-v) \le 2$ and $γ_g'(G) - γ_g'(G-v) \le 2$. Possibilities here are again realizable by connected graphs $G$ in almost all the cases, the exceptional values are treated similarly as in the edge-removal case.
Let $ρ(G)$ denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 <= g <=n. It is proved that $ρ(G) \leq \frac{n}{g}(m-n+1)$ and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.