In this paper it is proved that every $3$-connected planar graph contains a path on $3$ vertices each of which is of degree at most $15$ and a path on $4$ vertices each of which has degree at most $23$. Analogous results are stated for $3$-connected planar graphs of minimum degree $4$ and $5$. Moreover, for every pair of integers $n\ge 3$, $k\ge 4$ there is a $2$-connected planar graph such that every path on $n$ vertices in it has a vertex of degree $k$.

The eccentricity of a vertex $v$ of a connected graph $G$ is the distance from $v$ to a vertex farthest from $v$ in $G$. The center of $G$ is the subgraph of $G$ induced by the vertices having minimum eccentricity. For a vertex $v$ in a 2-edge-connected graph $G$, the edge-deleted eccentricity of $v$ is defined to be the maximum eccentricity of $v$ in $G-e$ over all edges $e$ of $G$. The edge-deleted center of $G$ is the subgraph induced by those vertices of $G$ having minimum edge-deleted eccentricity. The edge-deleted...

A graph $G$ with $p$ vertices and $q$ edges, vertex set $V\left(G\right)$ and edge set $E\left(G\right)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f,f^{+})$ where $f$ is a bijection from $V\left(G\right)$ onto $P$, $f^{+}$ is a bijection from $E\left(G\right)$ onto $Q$, $f^{+}\left((u,v)\right)=f\left(u\right)+f\left(v\right)$ for any $(u,v)\in E\left(G\right)$, $$Q=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}q\},\hfill & \text{if}q\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(q-1)\},\hfill & \text{if}q\text{is}\text{odd,}\hfill \end{array}\right.$$ and $$P=\left\{\begin{array}{cc}\{\pm 1,\cdots ,\pm \frac{1}{2}p\},\hfill & \text{if}p\text{is}\text{even,}\hfill \\ \{0,\pm 1,\cdots ,\pm \frac{1}{2}(p-1)\},\hfill & \text{if}p\text{is}\text{odd.}\hfill \end{array}\right.$$ We determine here families of unicyclic graphs that are super vertex-graceful.

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k)).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathrm{i}r\left(G\right)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order...