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A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to $m$. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths.

For a set ${\cal F}$ of graphs, an instance of the ${\cal F}$-{\sc free Sandwich Problem} is a pair $(G_1,G_2)$ consisting of two graphs $G_1$ and $G_2$ with the same vertex set such that $G_1$ is a subgraph of $G_2$, and the task is to determine an ${\cal F}$-free graph $G$ containing $G_1$ and contained in $G_2$, or to decide that such a graph does not exist. Initially motivated by the graph sandwich problem for trivially perfect graphs, which are the $\{ P_4,C_4\}$-free graphs, we study the complexity of the ${\cal F}$-{\sc free Sandwich Problem} for sets ${\cal F}$ containing two non-isomorphic graphs of order four. We show that if ${\cal F}$ is one of the sets $\left\{ {\rm diamond},K_4\right\}$, $\left\{ {\rm diamond},C_4\right\}$, $\left\{ {\rm diamond},{\rm paw}\right\}$, $\left\{ K_4,\overline{K_4}\right\}$, $\left\{ P_4,C_4\right\}$, $\left\{ P_4,\overline{\rm claw}\right\}$, $\left\{ P_4,\overline{\rm paw}\right\}$, $\left\{ P_4,\overline{\rm diamond}\right\}$, $\left\{ {\rm paw},C_4\right\}$, $\left\{ {\rm paw},{\rm claw}\right\}$, $\left\{ {\rm paw},\overline{{\rm claw}}\right\}$, $\left\{ {\rm paw},\overline{\rm paw}\right\}$, $\left\{ C_4,\overline{C_4}\right\}$, $\left\{ {\rm claw},\overline{{\rm claw}}\right\}$, and $\left\{ {\rm claw},\overline{C_4}\right\}$, then the ${\cal F}$-{\sc free Sandwich Problem} can be solved in polynomial time, and, if ${\cal F}$ is one of the sets $\left\{ C_4,K_4\right\}$, $\left\{ {\rm paw},K_4\right\}$, $\left\{ {\rm paw},\overline{K_4}\right\}$, $\left\{ {\rm paw},\overline{C_4}\right\}$, $\left\{ {\rm diamond},\overline{C_4}\right\}$, $\left\{ {\rm paw},\overline{\rm diamond}\right\}$, and $\left\{ {\rm diamond},\overline{\rm diamond}\right\}$, then the decision version of the ${\cal F}$-{\sc free Sandwich Problem} is NP-complete.

Important papers have appeared recently on the problem of indexing binary strings for jumbled pattern matching, and further lowering the time bounds in terms of the input size would now be a breakthrough with broad implications. We can still make progress on the problem, however, by considering other natural parameters. Badkobeh et al.\ (IPL, 2013) and Amir et al.\ (TCS, 2016) gave algorithms that index a binary string in $O (n + \rho^2 \log \rho)$ time, where $n$ is the length and $\rho$ is the number of runs, and Giaquinta and Grabowski (IPL, 2013) gave one that runs in $O (n + \rho^2)$ time. In this paper we propose a new and very simple algorithm that also runs in $O(n + \rho^2)$ time and can be extended either so that the index returns the position of a match (if there is one), or so that the algorithm uses only $O (n)$ bits of space.

For a graph $G$, let $\gamma_R(G)$ and $\gamma_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $\gamma_{r2}(G)\leq \gamma_R(G)\leq \frac{3}{2}\gamma_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $\gamma_R(G)-\gamma_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs efficiently. We show that for every fixed non-negative integer $k$, the recognition of the connected $K_4$-free graphs $G$ with $\gamma_R(G)-\gamma_{r2}(G)=k$ is NP-hard, which implies that there is most likely no good characterization of these graphs. We characterize the graphs $G$ such that $\gamma_{r2}(H)=\gamma_R(H)$ for every induced subgraph $H$ of $G$, and collect several properties of the graphs $G$ with $\gamma_R(G)=\frac{3}{2}\gamma_{r2}(G)$.

For a maximal outerplanar graph $G$ of order $n$ at least $3$, Matheson and Tarjan showed that $G$ has domination number at most $n/3$. Similarly, for a maximal outerplanar graph $G$ of order $n$ at least $5$, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that $G$ has total domination number at most $2n/5$ unless $G$ is isomorphic to one of two exceptional graphs of order $12$. We present a unified proof of a common generalization of these two results. For every positive integer $k$, we specify a set ${\cal H}_k$ of graphs of order at least $4k+4$ and at most $4k^2-2k$ such that every maximal outerplanar graph $G$ of order $n$ at least $2k+1$ that does not belong to ${\cal H}_k$ has a dominating set $D$ of order at most $\lfloor\frac{kn}{2k+1}\rfloor$ such that every component of the subgraph $G[D]$ of $G$ induced by $D$ has order at least $k$.

For a graph $G$, let $\gamma_{r2}(G)$ and $\gamma_R(G)$ denote the $2$-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) proved $\gamma_{r2}(G)+\gamma_R(G)\leq \frac{6}{4}n(G)$ for a connected graph $G$ of order $n(G)$ at least $3$. Furthermore, they conjectured $\gamma_{r2}(G)+\gamma_R(G)\leq \frac{4}{3}n(G)$ for a connected graph $G$ of minimum degree at least $2$ that is distinct from $C_5$. We characterize all extremal graphs for their inequality and prove their conjecture.

We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating function. Our characterization is based on five simple extension operations, and reveals several structural properties of these trees.

Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27-32) we prove $\gamma_{r2}(G)\leq 2\gamma_r(G)$ for every graph $G$, where $\gamma_{r2}(G)$ and $\gamma_r(G)$ denote the $2$-rainbow domination number and the weak Roman domination number of $G$, respectively. We characterize the extremal graphs for this inequality that are $\{ K_4,K_4-e\}$-free, and show that the recognition of the $K_5$-free extremal graphs is NP-hard.

Biclique-colouring is a colouring of the vertices of a graph in such a way that no maximal complete bipartite subgraph with at least one edge is monochromatic. We show that it is coNP-complete to check whether a given function that associates a colour to each vertex is a biclique-colouring, a result that justifies the search for structured classes where the biclique-colouring problem could be efficiently solved. We consider biclique-colouring restricted to powers of paths and powers of cycles. We determine the biclique-chromatic number of powers of paths and powers of cycles. The biclique-chromatic number of a power of a path P_{n}^{k} is max(2k + 2 - n, 2) if n >= k + 1 and exactly n otherwise. The biclique-chromatic number of a power of a cycle C_n^k is at most 3 if n >= 2k + 2 and exactly n otherwise; we additionally determine the powers of cycles that are 2-biclique-colourable. All proofs are algorithmic and provide polynomial-time biclique-colouring algorithms for graphs in the investigated classes., Comment: 21 pages, 19 distinct figures. An extended abstract published in: Proceedings of Cologne Twente Workshop (CTW) 2012, pp. 134--138